fourier-reflexive partitions and macwilliams identities for additive codes

Des. Codes Cryptogr.DOI 10.1007/s10623-014-9940-x

Fourier-reflexive partitions and MacWilliams identitiesfor additive codes

Heide Gluesing-Luerssen

Received: 5 April 2013 / Revised: 3 February 2014 / Accepted: 4 February 2014© Springer Science+Business Media New York 2014

Abstract A partition of a finite abelian group gives rise to a dual partition on the charactergroup via the Fourier transform. Properties of this dualization are investigated, and a con-venient test is given for when the bidual partition coincides with the primal partition. Suchpartitions permit MacWilliams identities for the partition enumerators of additive codes. It isshown that dualization commutes with taking products and symmetrized products of parti-tions on cartesian powers of the given group. After translating the results to Frobenius rings,which are identified with their character module, the approach is applied to partitions thatarise from poset structures

Keywords Weight enumerators · Partitions · Dualization · Poset structures

Mathematics Subject Classification 94B05 · 94B99 · 16L60

1 Introduction

MacWilliams identities relate properties of a code, in form of an enumerator or distribution,to properties of the dual code, and this relation is made precise by a concrete transformation.Such identities form an important tool for the theory of self-dual codes and also help to derivelinear programming bounds for codes, see, e.g., [3,6]. More engineering-oriented interest inMacWilliams identities can be found in [10,31,46].

The most famous MacWilliams identity is the one for the Hamming weight enumeratorfor codes over fields derived by MacWilliams [32]. A generalization to the symmetrized Leeweight appeared in the monograph by MacWilliams and Sloane [33].

Communicated by V. A. Zinoviev.

H. Gluesing-Luerssen (B)Department of Mathematics, University of Kentucky, 715 Patterson Office Tower,Lexington, KY 40506-0027, USAe-mail: [email protected]

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H. Gluesing-Luerssen

Starting in the 80s, these results were generalized further to codes over additive groupsor finite Frobenius rings and to other weight functions by Delsarte [6], Klemm [26,27],Nechaev and Kuzmin [34,35], Wood [45], and Honold and Landjev [21]; more literature willbe mentioned at the end of Sects. 3 and 5. The area of codes over rings gained a lot of attentionafter discovering the relevance of the Lee weight on Z4 for understanding the formal dualityof the binary non-linear Kerdock and Preparata codes by Hammons et al. [16]. In the realmof MacWilliams identities, the Frobenius property of the ring is essential as it guarantees thatthe character-theoretic annihilator of a code can be identified with the standard dual.

The most general approach was taken by Delsarte [6] with the aid of association schemes;see also Camion [4] and Delsarte and Levenshtein [8]. This approach allows to also dealwith non-linear codes and their distance distributions (again, with respect to various distancefunctions). Unfortunately, Delsarte’s results have not gained the attention they deserve, whichmay be due to the extended machinery of association schemes and their related Bose–Mesneralgebras. Many MacWilliams identities that form just special cases of Delsarte’s approach,have been proven independently afterwards.

In the 90s, Zinoviev and Ericson [47] revived the fundamental ideas of Delsarte by study-ing partitions of the ambient space, thus classifying the words according to a pre-specifiedproperty (e.g. the Hamming weight). In [47], they focused on additive codes and studiedthe case where the enumerators of the code and its dual with respect to the same partitionobey a MacWilliams identity. This led to the notion of a Fourier partition, reminding of thefact that the space of indicator functions of the partition sets is invariant under the Fouriertransform. In [48] the same authors extended their ideas to admissible pairs; these are par-tition pairs (P,Q) for which the Fourier transform induces an isomorphism between thetwo associated spaces of indicator functions. For such pairs the P-enumerator of a code andthe Q-enumerator of the dual code obey a MacWilliams identity. These pairs are exactlythose inducing an abelian association scheme on the second cartesian power of the group,thus relating this approach to Delsarte’s [6]. In [11] Forney generalized this result to discretesubgroups of locally compact abelian groups.

In [21] Honold and Landjev used a similar approach for linear codes over finite Frobeniusrings. By not using characters but rather a map with a certain homogeneous property, theirpartition pairs allowing a MacWilliams identity are in general characterized differently.

In this paper we will present an approach to MacWilliams identities solely based onpartitions of a group and their duals. This will also allow us to survey the vast literature onthe subject in more detail. We will make a careful distinction between the ambient space ofthe code and that of its dual: the code will be a subgroup of a given finite abelian group, and itsdual will reside in the character group. This is different from many of the settings above, wherethe code and its dual are contained in the same ambient space, mostly by identifying the groupwith its character group. However, since this identification is not canonical, many partitionproperties, most notably the dual of a partition, depend on the choice of the identification.Fortunately, this undesirable behavior does not occur for many standard situations, e.g., theHamming weight or the complete weight on any group.

When not identifying a group with its character group, the enumerator of a code and thatof the dual code refer to different partitions, as considered by Zinoviev and Ericson [48].Closely related is a notion introduced by Byrne et al. [3]. In the setting of codes over afinite Frobenius ring, identified with its character-module, they introduce the dual of a givenpartition with respect to the Fourier transform. This can be regarded as the one-sided analogueof the admissible pairs in [48] as now the two partitions need not be mutually dual. One oftheir motivations for dualizing a given partition appears to be the homogeneous weight. Thisparticular weight function induces in general a partition that is not a Fourier partition.

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Fourier-reflexive partitions and MacWilliams identities

In this paper we will draw ideas from many of the above mentioned papers and combinethem to a unified theory of partitions, partition enumerators and MacWilliams identitiesfor additive codes. Our starting point will be the dualization of a partition as introducedby Byrne et al. [3]. We will characterize when the bidual of a partition coincides with thegiven partition, and will call such partitions (Fourier-) reflexive. They correspond to theadmissible pairs in [48] and to abelian association schemes investigated in [6]. We willderive basic properties of dual partitions and present the resulting MacWilliams identityfor the associated partition enumerators. In Sect. 3 we will focus on product partitions andsymmetrized product partitions of a cartesian power of a finite abelian group. It will be shownthat these constructions commute with dualization.

All results will be derived directly from our definitions in order to show how they fallnaturally (and easily) into place. Whenever applicable, we will refer to analogous or closelyrelated results in the literature; they are often for a slightly different or more restrictedsetting.

In Sect. 4 we will briefly translate the setting to finite Frobenius rings, in which casewe will identify the ring with its character module. We will also relate our approach to thatin [21].

In Sect. 5 the approach will be applied to a particular recent area of MacWilliams identities,where the weight function is based on poset structures on the underlying index set {1, . . . , n}.This has been investigated by Kim and Oh [25] and Pinheiro and Firer [38]. The resultingweight generalizes the Rosenbloom–Tsfasman weight introduced in [39]. By studying theunderlying partitions we will generalize the MacWilliams identities to codes over groups.Once the desired partition duality is established, the identities are simply examples of thegeneral theory.

In a separate paper [13], we use our approach to investigate for which Frobenius rings thehomogeneous weight induces a reflexive partition.

2 Partitions and duality on finite abelian groups

In this section we review the basic material on character theory and the Fourier trans-form. We then go on and introduce the character-dual of a partition, discuss the impor-tant case where a partition coincides with its bidual, and present a general MacWilliamsidentity.

Throughout, let G be a non-trivial finite abelian (additive) group. Its character group isdefined as G := Hom(G,C∗) along with addition (χ1+χ2)(g) := χ1(g)χ2(g) for allχi ∈ G.The principal character of G is the trivial map χ ≡ 1 and denoted by ε. It is well-known [42]that the groups G and G are (non-canonically) isomorphic, and thus |G| = |G|. However, wecan and will identify the groups

G and G by mapping g ∈ G to the character of G that sendsχ ∈ G to χ(g). Thus g(χ) = χ(g). This suggests to write 〈χ, g〉 := χ(g), and thus we havethe identities 〈χ, g〉 = 〈g, χ〉 as well as 〈χ, g1 + g2〉 = 〈χ, g1〉〈χ, g2〉 and 〈χ1 + χ2, g〉 =〈χ1, g〉〈χ2, g〉. The finiteness of G implies |〈χ, g〉| = 1 and thus 〈χ, g〉−1 = 〈χ, g〉, thecomplex conjugate, for all χ ∈ G and g ∈ G.

It is easy to see [42, p. 171] that (G1 × · · · × Gn) = G1 × · · · × Gn for anygroups G1, . . . ,Gn , and where the character maps are given by

〈(χ1, . . . , χn), (g1, . . . , gn)〉 :=n

∏

i=1

〈χi , gi 〉 for all χi ∈ Gi , gi ∈ Gi . (2.1)

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A subgroup of G is called an (additive) code over G. For a code C ≤ G, the dual codeC⊥ ≤ G is defined as C⊥ = {χ ∈ G | 〈χ, h〉 = 1 for all h ∈ C}. Since C⊥ ∼= G/C, see [42,p. 198], we have |C⊥| · |C| = |G| and C⊥⊥ = C for any code C ≤ G.

The most important tool for duality theory are the orthogonality relations[44, Lemma (1.1.32)]

∑

g∈G

〈χ, g〉 = 0 if χ = ε and∑

g∈G

〈χ, g〉 = |G| if χ = ε. (2.2)

Let V be any complex vector space. The Fourier transform [42, p. 261] is the linearmap between the vector spaces V G and V G of maps from G (or G) to V given by f �−→f +, where f +(χ) = ∑

g∈G 〈χ, g〉 f (g). The identification of G with G and (2.2) lead to

f ++(g) = |G| f (−g), hence the Fourier transform is invertible. The Poisson summationformula [42, p. 199] states that for any code C ≤ G

∑

χ∈C⊥f +(χ) = |C⊥|

∑

h∈Cf (h). (2.3)

We now turn to partitions of G and their dual partitions of G. Let us fix the followingnotation. A partition P = (Pm)

Mm=1 of a set X , i.e., the sets Pm are non-empty, disjoint and

cover X , will mostly be written as P = P1 | P2 | . . . | PM . The sets of a given partition arecalled its blocks. We write |P| for the number of blocks in P . Two partitions P and Q ofa set X are called identical if |P| = |Q| and the blocks coincide after suitable indexing.Moreover, P is called finer than Q (or Q is coarser than P), written as P ≤ Q, if every blockof P is contained in a block of Q. Note that if P ≤ Q then |P| ≥ |Q|. Denote by ∼P theequivalence relation induced by P , thus, v∼Pv′ if v, v′ are in the same block of P .

The following notion of a dual partition will be central to our presentation. It has beenintroduced by Byrne et al. [3, p. 291] for Frobenius rings and goes back to the notion ofF-partitions as introduced by Zinoviev and Ericson in [47]. The special case of (Fourier-)reflexive partitions, defined below, corresponds to abelian association schemes as studiedby Delsarte [6], Camion [4], and others; see also [8,48]. If we identify the group G withits character group G, reflexive partitions turn out to be exactly the B-partitions studied byZinoviev and Ericson [48]. A similar approach to dualizing partitions (over Frobenius rings)is taken by Honold and Landjev [21]. Their main object are W-admissible pairs, which aremutually dual partitions on Rn , where R is Frobenius ring, and where duality is definedanalogously to below, but based on a certain homogeneous map instead of characters. Thisapproach works for Frobenius rings, where the existence of such a map is guaranteed. Laterin Remark 4.2 we will provide more details and discuss the relation to our partitions.

Definition 2.1 Let P = P1 | P2 | . . . | PM be a partition of G. The dual partition, denotedby P , is the partition of G defined by the equivalence relation

χ∼Pχ

′ :⇐⇒∑

g∈Pm

〈χ, g〉 =∑

g∈Pm

〈χ ′, g〉 for all m = 1, . . . ,M. (2.4)

Let P = Q1 | . . . | QL . The generalized Krawtchouk coefficients are defined as K�,m =∑

g∈Pm〈χ, g〉, where χ is any element in Ql . The matrix K = (K�,m) ∈ C

L×M is thegeneralized Krawtchouk matrix of (P, P). Finally, the partition P is called Fourier-reflexiveor simply reflexive if

P = P .

The relation to the Fourier transform will become clear in the proof of Theorem 2.4 below.The following properties are easy to verify.

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Remark 2.2 (a) The singleton {ε} is always a block of P , due to the orthogonality rela-tions (2.2). Moreover, if P consists of the single block G, then P = {ε} | G\{ε}.

(b) Let P = P1 | . . . | PM . Then P = −P = −P , where −P := −P1 | −P2 | . . . | −PM

and −Pm := {−g | g ∈ Pm}. This follows from 〈χ,−g〉 = 〈−χ, g〉 = 〈χ, g〉.(c) If P ≤ Q then P ≤ Q. This follows from the fact that each block of Q is a union of

blocks of P .

Example 2.3 (a) Consider G = Z6. Then G = {χa | a ∈ Z6}, where 〈χa, g〉 = ζ ag witha fixed primitive 6-th root of unity ζ ∈ C. We identify G and G via the isomorphisma �→ χa . Let P = 0 | 1, 3, 5 | 2, 4 (where we omit the parentheses). Computing for alla ∈ Z6 the sums

∑

g∈Pmζ ag one obtains P = 0 | 1, 2, 4, 5 | 3 and the Krawtchouk matrix

⎛

⎝

1 3 21 0 −11 −3 2

⎞

⎠ ,

where the rows and columns are indexed by the blocks of P and P in the given order. Inthe same way one can compute

P and obtains P = P . Hence P is reflexive.

(b) For the partition P = 0 | 1, 2 | 3, 4, 5 of Z6 one computes P = 0 | 1 | 2, 4 | 3 | 5, whileP consists of the singletons. Thus, P is not reflexive.

(c) This example will be needed later in Sect. 5. Let G = A1 × · · · × An , where Ai isa finite abelian group of order q ≥ 2 for all i . Let P be the partition of G given bythe Hamming weight, that is, Pm = {a ∈ G | wt(a) = m} for m = 0, . . . , n, andwhere wt(a1, . . . , an) = |{i | ai = 0}|. Similarly, let Q be the Hamming partitionof G = A1 × · · · × An , thus Q� = {χ ∈ G | wt(χ) = �}, where wt(χ1, . . . , χn) = |{i |χi = ε}|. Then

∑

a∈Pm〈χ, a〉 = K (n,q)

m (�) for each χ ∈ Q�,where

K (n,q)m (x) =

m∑

j=0

(−1) j (q − 1)m− j(

x

j

)(

n − x

m − j

)

(2.5)

is the Krawtchouk polynomial, see for instance [6, Theorem 4.1] or Lemma 2.6.2 in [23].Since K (n,q)

1 (�) = K (n,q)1 (�′) whenever � = �′, the partition Q is, as expected, the

dual of P , and thus K�,m = K (n,q)m (�) are the Krawtchouk coefficients of (P,Q). By

symmetry, P is reflexive. Identifying the isomorphic groups G and G we see that wemay call P self-dual, i.e., P = P (with respect to any isomorphism between G and G).

We now turn to a simple criterion for reflexivity. It states that P is reflexive if and onlyif the dual partition P has the same number of blocks as P . In the language of associationschemes, the following can also be found in [15, Theorem 10.1] (for partitions with blocksinvariant under taking inverses); see also [19, Fact V.2]. The counterpart for W-admissiblepairs is presented in [21, Theorem 12].

Theorem 2.4 Any partition P of G satisfies |P| ≤ |P| and P ≤ P . Moreover, P is reflexive

if and only if |P| = |P|.Proof Let P = P1 | P2 | . . . | PM and P = Q1 | Q2 | . . . | QL . Denote by ψm (resp. ξ�)the indicator function of the block Pm (resp. Q�). In the vector space C

G , consider them-dimensional subspace L(P) = 〈ψ1, . . . , ψM 〉C. By definition of the dual partition theFourier transforms ψ+

m , given by ψ+m (χ) = ∑

g∈Pm〈χ, g〉, are constant on each block Q�

of P . Hence

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L+(P) := 〈ψ+1 , . . . , ψ

+M 〉

C⊆ L(P) := 〈ξ1, . . . , ξL 〉C. (2.6)

Injectivity of the Fourier transform leads to |P| = M = dim L+(P) ≤ dim L(P) = L = |P|,which proves the first statement. Next, the inverse Fourier transform along with Remark 2.2(b)and (2.6) show that L(−P) = L++(P) ⊆ L+(P) = L+(−P). This implies L(P) ⊆L+(P) = 〈ξ+

0 , . . . , ξ+L 〉

C. In other words, the indicator functionsψm are linear combinations

of ξ+0 , . . . , ξ

+L . Since the Fourier transforms ξ+

� are constant on each block of the partitionP , the same is true for ψm . Thus g∼

P g′ implies ψm(g) = ψm(g′) for all m, and this means

g∼P g′. This showsP ≤ P .

The above also shows that if P is reflexive then |P| = |P|. Thus it remains to show thatif |P| = |P|, then P ≤

P . Let |P| = |P|. Then (2.6) yields L+(P) = L(P). The abovereasoning also implies L(P) = L+(P) = 〈ξ+

0 , . . . , ξ+L 〉

C, and this means that each ξ+

�

is a linear combination of ψ1, . . . , ψM . By definition of ξ+� this leads to the implication

g∼P g′ �⇒ g∼P g′ for all g, g′ ∈ G, and this concludes the proof. ��

As already hinted at, reflexive partitions are closely related to abelian association schemes.Indeed, with the above result and some straightforward, but lengthy computations one canshow that the partition P is reflexive if and only the partition R = R1 | R2 | . . . | RM ofG × G defined via (x, y) ∈ Rm ⇔ x − y ∈ Pm is an abelian association scheme. In thiscase, P gives rise to the dual association scheme. With the machinery of association schemes,this has already been established in [4, Corollary 4.51] as well as [48, Theorem 1] and goesback to [6, Sect. 2.6.1].

Let us return to the general situation of dualizing partitions. It is natural to ask whetherthe operations on the lattice of partitions of G (w.r.t. ≤) are respected by dualization. Recallthat the join, P ∨Q, of two partitions P and Q is defined as the finest partition that is coarserthan both P and Q, and the meet, P ∧ Q, is defined as coarsest partition that is finer thanboth P and Q. The following result can also be found in [4, Sect. 4.13] within the theoryof association schemes and in [21, Theorem 13] for W-admissible partitions over Frobeniusrings.

Proposition 2.5 If P and Q are reflexive partitions of G, then so isP∨Q and P ∨ Q = P∨Q.

The analogous statements for the meet P ∧ Q are in general not true as can be verifiedwith the reflexive partitions P = 0 | 1, 7 | 2, 6 | 3, 5 | 4 and Q = 0 | 1, 3 | 2, 6 | 4 | 5, 7 of Z8.

Proof By definition of the join, P ≤ P ∨ Q, and thus P ≤ P ∨ Q due to Remark 2.2(c).

Again by definition of the join, P∨ Q ≤ P ∨ Q. Taking duals we obtain P = P ≤ P ∨ Q and

thus P ∨Q ≤ P ∨ Q. Now Theorem 2.4 yields reflexivity. Applying this line of reasoning to

P ∨ Q, we obtain P ∨ Q = P ∨ Q ≥ P ∨ Q. All of this shows P ∨ Q = P ∨ Q, as desired. ��

The following class of partitions will be needed in Sect. 3. Recall that for any groupautomorphismρ ∈ Aut(G) the adjoint automorphismρT ∈ Aut(G) is defined by 〈ρT(χ), g〉 =〈χ, ρ(g)〉 for all χ ∈ G and g ∈ G. For H ≤ Aut(G) the adjoint group HT ≤ Aut(G) isdefined as HT = {ρT | ρ ∈ H}. As we show next, the partition given by the orbits of H isreflexive; see also [4, Lemma 4.63] for the analogue within association schemes and [21,Theorem 16] for W-admissibility over Frobenius rings.

Theorem 2.6 Let H ≤ Aut(G) and PH be the partition of G given by the orbits of H.Then PH is reflexive and PH = PHT .

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Proof Let χ∼PHTχ′, thus χ ′ = ρT(χ) for some ρ ∈ H . Moreover, fix g ∈ G and consider

the orbit P = {σ(g) | σ ∈ H}, which thus is a block of PH . Then∑

g∈P 〈χ, g〉 =∑

σ∈H 〈χ, σ (g)〉 = ∑

σ∈H 〈σT(χ), g〉 = ∑

σ∈H 〈σT(χ ′), g〉 = ∑

g∈P 〈χ ′, g〉. Thus,

χ∼PHχ ′. In the same way, g∼PH g′ implies g∼PHT

g′. Hence PHT ≤ PH and PH ≤ PHT ,

and Theorem 2.4 yields PH = PHT . Reflexivity is a consequence of HT T = H . ��

We close the section with a MacWilliams identity for partition enumerators of codes in Gand their dual codes in G. The result can also be found in [4, Theorem 4.72, Prop. 5.42], whereit has been derived with the aid of association schemes, in [11, p. 94] for discrete subgroups oflocally compact abelian groups, and in [21, Theorem 21] for W-admissible pairs over Frobe-nius rings. The connection between abelian association schemes and MacWilliams identitieswas observed already in [6, p. 88]. For the special case of self-dual partitions (i. e., P = Pwhen identifying G and G), the identity was also established in [47, Theorem 1], and forsubmodules of Rn , where R is a Frobenius ring identified with R, and where the partition Qarises from symmetrization of a partition on R, it appears also in [3, Theorem 2.11].

Let P = P1 | . . . | PM and Q = Q1 | . . . | QL be partitions of G and G, respectively, suchthat P = Q (note that Q is the primal partition and P its dual). Let K = (Km,�) ∈ C

M×L bethe Krawtchouk matrix of (Q,P). For a code C ≤ G and its dual C⊥ ≤ G define the partitionenumerators PEP,C ∈ C[X1, . . . , X M ] and PEQ,C⊥ ∈ C[Y1, . . . , YL ] as

PEP,C =M

∑

m=1

Am Xm, PEQ,C⊥ =L

∑

�=1

B�Y�,

where Am = |C ∩ Pm | and B� = |C⊥ ∩ Q�|. (2.7)

They carry the information about the number of codewords contained in each block of thepartitions.

The MacWilliams identity below is based on a transformation M of the enumerator PEP,Cresulting in the enumerator PEQ,C⊥ . Its well-definedness is guaranteed by the identity P = Q.In general, it is not possible to invert the operator and compute PEP,C from PEQ,C⊥ . Thisis only guaranteed if Q = P , i.e., if P and Q are both reflexive and thus mutually dual.Thus, not surprisingly, reflexive partitions provide a symmetric situation and form the mostappealing case.

We will make use of this identity later in Sect. 5.

Theorem 2.7 Define the MacWilliams transformationM : C[X1, . . . , X M ] −→ C[Y1, . . . ,

YL ] as the algebra homomorphism given by M(Xm) = ∑L�=1 Km,�Y� for m = 1, . . . ,M.

Then

PEQ,C⊥ = 1

|C|M(PEP,C). (2.8)

In other words, (B1, . . . , BL) = |C|−1(A1, . . . , AM )K , where K is the Krawtchouk matrix.

Proof Define the maps ψ : G → C[X1, . . . , X M ], g �→ Xm(g), where m(g) is the uniqueindex such that g ∈ Pm(g). Similarly, let ψ : G → C[Y1, . . . , YL ]χ �→ Y�(χ), where �(χ) issuch that χ ∈ Q�(χ). Then PEP,C = ∑

g∈C Xm(g) and PEQ,C⊥ = ∑

χ∈C⊥ Y�(χ). Using thedefinition of the Krawtchouk coefficients we obtain for g ∈ Pm

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M(Xm) =L

∑

�=1

Km,�Y� =L

∑

�=1

∑

χ∈Q�

〈χ, g〉ψ(χ) =∑

χ∈G

〈χ, g〉ψ(χ) = ψ+(g),

where ψ+ is the Fourier transform. The Poisson formula (2.3) applied to the map ψ yields

M(PEP,C) =∑

g∈CM(Xm(g)) =

∑

g∈Cψ+(g) = |C|

∑

χ∈C⊥ψ(χ) = |C|PEQ,C⊥ .

��It is easy to see that the MacWilliams identity can be generalized to the following situation.

Suppose P = Q and P ′ ≤ P and Q′ ≥ Q, i.e., P ′ is finer than P and Q′ is coarser than Q. Thenone obtains a MacWilliams transformation of PEP ′,C resulting in PEQ′,C⊥ . For symmetrizedpartitions on Rn , where R is a Frobenius ring, this can be found in [3, Theorem 2.11].

3 Induced partitions

In this section we turn to cartesian products of groups. We present two specific constructionsof induced partitions and the resulting MacWilliams identities. They cover many of theknown MacWilliams identities in coding theory. Product partitions, as defined below, werediscussed before in [48, Sect. 7] and [4, Sect. 4.10], while symmetrized partitions appearedin [6, Sect. 2.5] and [4, Sect. 4.11]. See also [21, Sect. 3] for the analogues for W-admissiblepairs of partitions.

As before, any finite abelian group is supposed to be non-trivial.

Definition 3.1 Let G = G1 × · · · × Gn , where each Gi is a finite abelian group, and letPi = Pi,1 | Pi,2 | . . . | Pi,Mi be a partition of Gi for i = 1, . . . , n. On G define the inducedproduct partition as P1 × · · · × Pn := (P1,m1 × · · · × Pn,mn )(m1,...,mn)∈[M1]×···×[Mn ], where[Mi ] := {1, . . . ,Mi }. If Gi = G and Pi = P for all i , then P × · · · × P is written as Pn .

Definition 3.2 Let G = Gn and P = P1 | P2 | . . . | PM be a partition of the finite abeliangroup G. On G we define the equivalence relation (g1, . . . , gn)∼(g′

1, . . . , g′n) if |{i | gi ∈

Pm}| = |{i | g′i ∈ Pm}| for all m = 1, . . . ,M . The associated partition on G is denoted by

Pnsym and called the induced symmetrized partition.

Note that each block of the product partition P1×· · ·×Pn consists of all g ∈ G1×· · ·×Gn

for which each entry gi is contained in a prescribed block of Pi . Obviously, P1 × · · · ×Pn has M1 · . . . · Mn blocks. In particular, |Pn | = Mn . On the other hand, the blocks ofthe symmetrized partition Pn

sym collect all g ∈ Gn that have the same number of entries(disregarding position) in any given block of P . The number of block of Pn

sym equals the

number of weak M-partitions of n, thus |Pnsym| = (n+M−1

M−1

)

, see [41, p. 15].One may note that the symmetrized partition is the join Pn ∨ Q, where Q is the orbit

partition of the subgroup Sym(n) ≤ Aut(Gn). We will make use of this fact in the proofof Theorem 3.3(b) below. This construction is an instance of a fusion as defined in [21,Definitions 15, 18].

As a simple example, if P = {0} ∣

∣ G\{0}, then Pn partitions the elements (g1, . . . , gn) ∈Gn according to their support, whereas Pn

sym classifies them with respect to their Hammingweight.

As we show next, dualization commutes with the above constructions under the ratherweak condition that {0} is a block of the given partitions. Let us first consider an (extreme)

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example illustrating the necessity of this condition. Suppose P consists of the single block G.Then both Pn and Pn

sym consist of the single block Gn and Pn = Pnsym = {ε} ∣

∣ Gn\{ε}, as

we saw in Remark 2.2(a); recall from (2.1) the identification of Gn with Gn . In particular,P = {ε} ∣

∣ G\{ε}. Therefore, P n consists of all n-fold product sets with factors {ε} and G\{ε},and P n

sym is the Hamming partition of Gn . All of this shows P n� Pn and P n

sym � Pnsym.

The statements in the following theorem pertaining to reflexive partitions appears alsoin [48, Theorem 4], [6, Sect. 2.5] and [4, Theorems 4.86, 4.97]. The counterparts for W-admissible partitions can be found in [21, Theorem 14, Corollary 17].

Theorem 3.3 (a) For all i = 1, . . . , n let Pi be a partition of Gi containing the block {0}.Let Q = P1 × · · · × Pn. Then Q = P1 × · · · × Pn. As a consequence, if all Pi arereflexive, then so is P1 × · · · × Pn.

(b) Let P be a partition of G such that {0} is a block of P . Then Pnsym = P n

sym. As a

consequence, if P is reflexive then so is Pnsym.

Proof (a) Write Pi be as in Definition 3.1 and Pi = Qi,1 | Qi,2 | . . . | Qi,Li . With the aidof (2.1) we compute for all χ ∈ G

∑

g∈P1,m1 ×···×Pn,mn

〈χ, g〉 =∑

g∈P1,m1 ×···×Pn,mn

n∏

i=1

〈χi , gi 〉 =n

∏

i=1

∑

g∈Pi,mi

〈χi , g〉. (3.1)

This shows immediately that if χ∼P1×···×Pn

χ ′ then χ∼Qχ

′. Considering (3.1) forall product partition blocks where Pi,mi = {0} for all but one i , establishes the converse.(b)1 Consider the subgroup Sym(n) ≤ Aut(Gn) consisting of the permutations on Gn ,and let Q = PSym(n) be the induced orbit partition on Gn , see Theorem 2.6. Due to thattheorem Q is the orbit partition of the symmetry group on Gn . By the very definition ofsymmetrized partitions, Pn

sym equals the join Pn ∨Q. Now Proposition 2.5 and part (a) imply

Pnsym = Pn ∨ Q = Pn ∨ Q = P n ∨ Q = P n

sym. ��Now we derive the MacWilliams identities for the induced product and symmetrized

partitions of Gn . We restrict ourselves to the reflexive case. The general case can easily bedealt with, but as in Theorem 2.7 requires slightly more notation and the careful choice of thepartition and its dual. For the reflexive case, analogous identities in the language of abelianassociation schemes appear in [4, Theorems 5.46, 5.51], while for invariant W-admissiblepairs a counterpart is to be found in [21, Theorem 22].

For the first result we assume the situation of Definition 3.1. Write Q = P1 × · · · × Pn .Let the dual partitions be Pi = Qi,1 | . . . | Qi,Mi , and thus the blocks of Q are given byQ� = Q1,l1 ×· · ·× Qn,ln for all index combinations � = (l1, . . . , ln). For g ∈ Gi we denoteby m(g) ∈ {1, . . . ,Mi } the index of the block Pi,m(g) containing g, and similarly for χ ∈ Gi

let χ ∈ Qi,m(χ).

Theorem 3.4 Let all partitions Pi be reflexive, thus Q is reflexive as well, and let K (i) ∈C

Mi ×Mi be the Krawtchouk matrix of the pair (Pi ,Pi ). For a code C ≤ G we define theproduct partition enumerator as PEQ,C := ∑

g∈C∏n

i=1 Xi,m(gi ) ∈ W := C[Xi,m | i =1, . . . , n, m = 1, . . . ,Mi ]. Hence the coefficient of

∏ni=1 Xi,mi equals the cardinality of

C ∩ (P1,m1 × · · · × Pn,mn ). The enumerator satisfies the MacWilliams identity

PEQ,C⊥ = 1

|C|M′(PEQ,C), (3.2)

1 The author is grateful to the anonymous reviewer for suggesting this line of proof.

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where M′ : W −→ W is the algebra homomorphism given by Xi,m �→ ∑Mi�=1 K (i)

m,�Xi,� for

i = 1, . . . , n and m = 1, . . . ,Mi . Defining Xi := (Xi,1, . . . , Xi,Mi )T, we may write (3.2)

also as

PEQ,C⊥(Xi | i = 1, . . . , n) = |C|−1 PEQ,C(K (i)Xi | i = 1, . . . , n). (3.3)

Proof Define the maps ψ : G → W, g �→ ∏ni=1 Xi,m(gi ) and ψ : G → W, χ �→

∏ni=1 Xi,m(χi ). Then PEQ,C = ∑

g∈C ψ(g) and similarly PEQ,C⊥ = ∑

χ∈C⊥ ψ(χ). By

the definition of the Krawtchouk coefficients, K (i)m,l = ∑

χ∈Qi,l〈χ, g〉 for any g ∈ Pi,m .

Using (2.1) we obtain for g ∈ P1,m1 × · · · × Pn,mn

ψ+(g) =∑

χ∈G〈χ, g〉ψ(χ) =

∑

�=(�1,...,�n)

∑

χ∈Q�

n∏

i=1

〈χi , gi 〉Xi,�i

=n

∏

i=1

Mi∑

�=1

∑

χi ∈Qi,�

〈χi , gi 〉Xi,� = M′(

n∏

i=1

Xi,mi

)

.

The Poisson summation formula (2.3) yields the desired result. ��In the same way we can derive a MacWilliams identity for the induced symmetrized

partitions of Gn . As before, let m(g) be the unique index such that g ∈ Pm(g).

Theorem 3.5 Let P = P1 | . . . | PM be a reflexive partition of G and K ∈ CM×M be the

Krawtchouk matrix of (P,P). For a code C ≤ Gn the symmetrized partition enumeratorof C with respect to P is defined as PEPn

sym,C := ∑

g∈C∏n

t=1 Ym(gt ) ∈ V := C[Ym |m = 1, . . . ,M]. The coefficient of the monomial

∏Mm=1 Y sm

m counts all g ∈ C for which|{i | gi ∈ Pm}| = sm for all m = 1, . . . ,M. The enumerator satisfies the MacWilliamsidentity

PEPn

sym,C⊥ = 1

|C|M′′(PEPn

sym,C), (3.4)

where M′′ : V −→ V is the algebra homomorphism given by Ym �→ ∑M�=1 Km,�Y� for

m = 1, . . . ,M. (3.4) may also be written as PEPn

sym,C⊥(Y) = |C|−1PEPnsym,C

(

K Y)

, where

Y = (Y1, . . . , YM )T.

Proof We make use of Theorem 3.4, applied to the product partition Pn and its dual. Considerthe polynomial ring W = C[Xi,m | i = 1, . . . , n, m = 1, . . . ,M] and the substitution homo-morphism ρ : W −→ V , Xt,m �−→ Ym . Theorem 3.4 yields PE

Pnsym,C⊥ = ρ(PE

Pn ,C⊥) =|C|−1ρ ◦ M′(PEPn ,C). Note that ρ ◦ M′(Xt,m(g)) = ∑

χ∈G 〈χ, g〉Ym(χ) = M′′(Ym(g)) =M′′ ◦ ρ(Xt,m(g)). Hence ρ ◦ M′(PEPn ,C) = M′′ ◦ ρ(PEPn ,C) = M′′(PEPn

sym,C), and thisconcludes the proof. ��

The last two theorems cover an abundance of MacWilliams identities from the litera-ture: the identities for the Hamming weight, the complete weight [4,27,33,45], the exactweight [4,33], the symmetrized Lee weight [16,26,45] are all instances of Theorem 3.5 forthe symmetrized partition enumerator, and so are many other cases. The only work left isthe explicit computation of the Krawtchouk coefficients in each concrete case. For the justmentioned examples, this can be done straightforwardly; see also [14, Sect. 4], where thishas been carried out in detail. Most of the literature just mentioned deals with codes over

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fields or Frobenius rings, in which case the character dual is identified with the ring, and thedual code is the classical dual with respect to the inner product. We will discuss this relationin the next section.

While the above cases are well known, we wish to touch upon a different class of lesserknown identities. That is those of split weight enumerators, where for instance, the Hammingweight is considered separately on various components of a given vector, or the Hammingweight is considered on one part and the symmetrized Lee weight on the other one. Allthese cases are based on product partitions and thus instances of Theorem 3.4. The resultingMacWilliams identity for the split Hamming weight enumerator was derived by MacWilliamsand Sloane for codes over the binary field in [33, Chap. 5, Eq. (52)] and by Simonis [40,Eq. (3′)] for arbitrary fields and where the codewords are divided into t blocks of coordinates.Similar identities can be found in [10] by El-Khamy and McEliece and by Lu et al. [31].

More recently, Yang [46] introduced the second-order weight distribution of codes overfields. They play an important role in computing the second moments, which in turn havepromising applications to random coding. In the language of this paper, the second-orderweight distribution of a product code C × C′ ⊆ F

n × Fn is the partition enumerator of C × C′

considered as a code of length n over the Frobenius ring R := F × F (see also next section),with respect to a particular self-dual partition of Rn (i.e., the partition coincides with its leftand right-dual in the sense of Definition 4.1). As a consequence, the MacWilliams identityin [46, Theorem 2.12] is yet another instance of Theorem 3.4. This has also been pointed outin [46, Remark 2.13].

The above examples are all based on a product or symmetrized partition. But in the litera-ture also MacWilliams identities have been derived that are not based on induced partitions.Most notably, in [7] Delsarte established a MacWilliams identity for the rank partition onF

s×n and also computed the Krawtchouk coefficients. This partition is an orbit partition asin Theorem 2.6 and thus reflexive (it is even self-dual if one identifies F

s×n with its charactergroup, see Sect. 4). The identity in [7] thus forms an instance of Theorem 2.7, which is basedon general partitions on Rn . However, the computation of the Krawtchouk coefficients isquite technical in this case. Another application of Theorem 2.7 will be given in Sect. 5 forposet weights on product groups. At the end of that section we will also apply Theorems 3.4and 3.5 to induced poset weights on Rs×n .

4 Duality for Frobenius rings

In this section we focus on the case where the group G is the additive group of a finite ring.The results of this section are not new, and the goal is rather to carefully reconcile duality inthe group setting with that for codes over rings. In the literature concerning duality of codesover a ring R, the dual of a code (i.e., submodule) in Rn is usually defined as the orthogonalsubmodule in Rn , similar to the classical case of codes over a field. As we will see, thisamounts to the same as the group theoretic dual if the ring R is Frobenius. Indeed, in that casethe character group Rn is isomorphic to Rn as an R-module. As a consequence, the previousresults on partitions and their duals apply. We will also discuss that the identification of Rn

with Rn is not canonical and therefore the dual of a partition depends on this identification.Let R be a finite ring with identity. Its group of units is denoted by R∗. For any n ∈ N we

denote by Rn the character group of the additive group (Rn,+). This group can be endowedwith an R-R-bimodule structure via the left and right scalar multiplications

(rχ)(v) = χ(vr) and (χr)(v) = χ(rv) for all r ∈ R and v ∈ Rn . (4.1)

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Recall that R is a Frobenius ring if R/rad(R) ∼= soc(R R) as left R-modules, where rad(R)is the Jacobson radical of R and soc(R R) is the left socle of R. It follows from [29], [17,Theorem 1], [18, p. 409], and [45, Sect. 3] that R is Frobenius if and only if R R and R R areisomorphic. In other words, R is Frobenius if and only if there exists a character χ ∈ R suchthat

R R −→R R, r �−→ rχ, (4.2)

is an isomorphism. It has been shown in [45, Theorem 4.3] that (4.2) implies the right R-isomorphism RR −→ RR, r �−→ χr . Such a character χ is called a generating characterof R. It is a consequence of [45, Proposition 5.1] that any two generating characters χ, χ ′differ by a unit, i.e., χ ′ = uχ and χ ′ = χu′ for some u, u′ ∈ R∗.

Examples of finite Frobenius rings are finite fields, integer residue rings ZN , finite chainrings, matrix rings Rn×n and finite group rings R[G] over Frobenius rings R. Direct productsof finite Frobenius rings are Frobenius. A simple example of a non-Frobenius ring is thecommutative ring R = F2[x, y]/(x2, y2, xy), see [5, Example 3.2], but there exist alsonon-commutative non-Frobenius rings.

From now on, let R be a Frobenius ring with generating character χ . Using (4.1) and (2.1)it is easy to see that (4.2) and its right analogue extend to isomorphisms

αl : R(Rn) −→ R(Rn), x �−→ χ(− · x) ,

αr : (Rn)R −→ (Rn)R, x �−→ χ(x · −), (4.3)

where x · y denotes the standard inner (dot) product on Rn .Let C be a subgroup of Rn . As in Sect. 2, the dual group is C⊥ = {ψ ∈ Rn |

ψ(v)= 1for all v ∈ C}. It is immediate that if C is a left (resp. right) submodule of Rn

then C⊥ is a right (resp. left) R-submodule of Rn . Using that the kernel of a generatingcharacter does not contain any non-zero one-sided ideals [5, Corollary 3.6], one obtains (seealso [45, Theorem 7.7])

C⊥,r := α−1l (C⊥) = {v ∈ Rn | w · v = 0 for all w ∈ C} if C is a left R-module,

C⊥,l := α−1r (C⊥) = {v ∈ Rn | v · w = 0 for all w ∈ C} if C is a right R-module. (4.4)

Evidently, C⊥,r is a right R-module and C⊥,l is a left one. Thus, αl and αr act only asgroup isomorphisms.

With the above we recover the classical duals as they are commonly considered in thetheory of codes over rings or fields. It is worth mentioning that also the case of Fq -linear codesin F

nqt and their duals under the trace inner product, as considered for instance in the recent

paper [22], are covered by the above considerations. Indeed, if ψ is a generating characterof Fq , then the map χ := ψ ◦ traceqt ,q is a generating character of Fqt . With this χ oneobtains C⊥,r = α−1

l (C⊥) = {v ∈ Rn | traceqt ,q(w · v) = 0 for all w ∈ C}, which is thefamiliar dual under the trace inner product.

Let us return to the general case of (4.3). It follows from |C||C⊥| = |Rn | (see Sect. 2)that (C⊥,r )⊥,l = C for any left submodule C ⊆ Rn . If n = 1 this identity is known as thedouble annihilator property for left ideals, see, e.g., [28, Theorem 15.1]. This property is ingeneral not true if R is not Frobenius as can easily be seen using the ideal C = (x) in thering R = F2[x, y]/(x2, y2, xy). In this case, (x)⊥,r = (x, y) and ((x)⊥,r )⊥,l = (x, y) = (x)(thereby proving that R is not Frobenius).

With the aid of the isomorphisms αl and αr we may define the dual of a partition of Rn asa partition of Rn . However, since αl and αr depend on the choice of the generating character,

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the same is true in general for the dual partition. We cast the following definition and discussthe dependence on χ afterwards; see also [1, Definition 4.3].

Definition 4.1 Let P = (Pm)Mm=1 be a partition of Rn . The left and right χ-dual partitions

of P , denoted by P [χ,l]and P [χ,r]

, are defined as α−1l (P) and α−1

r (P), respectively, andwhere P is the dual in the sense of Definition 2.1. In other words, they are given by theequivalence relations

v∼P [χ,l] v′ :⇐⇒ ∑

w∈Pmχ(w · v) = ∑

w∈Pmχ(w · v′) for all m = 1, . . . ,M, (4.5)

v∼P [χ,r] v′ :⇐⇒ ∑

w∈Pmχ(v · w) = ∑

w∈Pmχ(v′ · w) for all m = 1, . . . ,M. (4.6)

It is not hard to find examples of a ring and a partition P for which P [χ,l] = P [χ,r]for any

generating character (take, e.g., the ring in [45, Example 1.4(iii)]).Both duals of a partition depend in general on the choice of the generating character. This

can be described as follows. Suppose χ and χ are both generating characters of R. Then χ =uχ for some u ∈ R∗, and (4.6) along with (4.1) shows that v∼

P [χ ,l] v′ ⇐⇒ vu∼P [χ,l] v′u

for any v, v′ ∈ Rn . Hence P [χ,l] = P [χ ,l]u, where the latter denotes the partition obtained

by multiplying each block of P [χ ,l]by u on the right. This shows that the left dual partitions

differ by a right unit factor. Using that χ = χu′ for some u′ ∈ R∗, we also observe from (4.6)that if P is invariant under left multiplication by units, then its left dual partition does notdepend on the choice of the generating character. Analogous statements are true for the rightdual.

It is not hard to see [1, Proposition 4.4] that the bidual partition does not depend on thegenerating character. Precisely,

p [χ,l][χ,r ] = p = p [χ,r ][χ,l]

for any generating character χ , and where p is the bidual partition in the group sense of (2.4).All of this allows to apply the previously derived MacWilliams identities for partition

enumerators to codes over Frobenius rings and their dual defined via the standard innerproduct.

We close this section with comparing our approach to that of Honold and Landjev [21];see also the predecessors [30] for the case where R is a finite field, and [20], where R is afinite chain ring.

Remark 4.2 In [21] Honold and Landjev study codes C and their duals C⊥,r in R Rn , where Ris a Frobenius ring. Dualization of partitions is defined as in Definition 4.1, but where insteadof χ a map ω : R → Q is used that satisfies ω(0) = 1, ω(ur) = ω(r) for all r ∈ R andu ∈ R∗ and

∑

r∈I ω(r) = 0 for each nonzero left ideal of R. For Frobenius rings such amap exists and is unique. It is given by the average of the generating characters; see [18,Theorems 1 and 2]. A pair of partitions (P,Q) is called W-admissible if P and Q are mutuallydual in the ω-sense. This approach relates to ours as follows. Firstly, one can show that if Pis a reflexive and left invariant partition (i.e., each block is invariant under left multiplicationby units), then (P, P [χ,l]

) is W-admissible. Furthermore, if (P,Q) is W-admissible and Pis left invariant and Q is right-invariant, then Q = P [χ,l]

and P is reflexive. However, onecan construct non-invariant W-admissible partitions that are not reflexive and vice versa. Forinstance, the partition P = 0 | 1, α |α2 of F4 is reflexive and even self-dual, i.e., P = P [χ,l]

,with respect to the character χ given by χ(0) = χ(1) = 1 and χ(α) = χ(α2) = −1. Hence

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the partition P2sym of F

24 is self-dual. But there exists no partition Q of F

24 such that (P2

sym,Q)is W-admissible. In this sense our approach and the one in [21] do not cover entirely thesame situations.

5 Poset structures on G1 × · · · × Gn

Let Gi be non-trivial finite abelian groups for i = 1, . . . , n. In this section we consider aweight for codes in G := G1 × · · · × Gn that derives from a prescribed poset structure onthe coordinate set [n] := {1, . . . , n}. This was introduced by Brualdi et al. [2] for codesover fields (i.e., codes in F

n), where it generalizes the Hamming weight as well as theNiederreiter–Rosenbloom–Tsfasman weight. The latter was introduced by Rosenbloom andTsfasman in [39]. Even though these weights originate from a combinatorial problem posedby Niederreiter [36], they find interesting applications for transmissions over parallel slowfading channels, see for instance [12,37,39].

In [9] Dougherty and Skriganov established a MacWilliams identity for the Rosenbloom–Tsfasman weight and codes over fields. In [25] Kim and Oh generalize this result to posetweights induced by hierarchical posets, and they also show that these posets are the onlyones that allow a MacWilliams identity. Pinheiro and Firer [38] generalized this even furtherto block poset structures. In all these cases the dual weight is induced by the dual poset.

In this section, we will recover these MacWilliams identities and generalize them to codesover groups. We will do so by establishing that the partitions induced by a hierarchical posetweight and the dual poset weight are mutually dual in the sense of Definition 2.1. Thus theMacWilliams identity is simply an instance of Theorem 2.7.

Let ≤ be a partial order on [n], thus P := ([n],≤) is a poset. We denote by max(P) (resp.min(P)) the set of all maximal (resp. minimal) elements of P. A subset S ⊆ [n] is called anideal if i ∈ S and j ≤ i implies j ∈ S. Denote by 〈S] the smallest ideal containing the set S.

A poset P = ([n],≤) induces the poset weight on G defined as

wtP(g) = ∣

∣〈supp(g)]∣∣, (5.1)

where, as usual, supp(g) := {i | gi = 0} denotes the support of g = (g1, . . . , gn) ∈ G. Theweight induces a metric on G (see [2, Lemma 1.1] for F

n). More interesting to us, it gives riseto a partition PP = P0 | . . . | Pn of G via Pm = {g ∈ G | wtP(g) = m} for m = 0, . . . , n. Itis easy to see that the blocks Pm are nonempty for each m and thus |PP| = n + 1.

A particular role is played by hierarchical posets. For the terminology we follow [25].

Definition 5.1 A poset P = ([n],≤) is called hierarchical if there exists a partition [n] =⋃t

i=1· �i such that for all l,m ∈ [n] we have l < m if and only if l ∈ �i1 , m ∈ �i2 for some

i1 < i2 (where i1 < i2 refers to the natural order in N). In other words, for all i1 < i2 everyelement in �i1 is less than every element in �i2 , and no other two distinct elements in [n] arecomparable. We call �i the i-th level of P. The hierarchical poset is completely determined(up to order-isomorphism) by the data (n1, . . . , nt ), where ni = |�i |, and is denoted byH(n; n1, . . . , nt ).

There are two extremes cases. First, the hierarchical poset H(n; n) is called an anti-chainon [n] because any two distinct elements are incomparable. In this case, 〈supp(v)] = supp(v)for all g ∈ G, and wtP is simply the Hamming weight on G. Secondly, a poset P = ([n],≤)is a chain if ≤ is a total order. Thus, P is a chain if and only if P = H(n; 1, . . . , 1). Assumingwithout loss of generality that 1 < 2 < · · · < n, we observe that wtP(g) = max{i | gi = 0},which on F

n is known as the Rosenbloom–Tsfasman weight.

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The dual of the poset P is defined as the poset P = ([n],≥) where x ≥ y :⇐⇒ y ≤ x . Itis obvious that if P is a hierarchical poset with t levels, then so is P, and the �-th level of Pis the (t + 1 − �)-th level of P.

For an inductive argument later on, it is beneficial to consider a particular situation before-hand.

Remark 5.2 Let P = ([n],≤) be a poset, and let max(P) = {l + 1, . . . , n}, which then isalso min(P). Suppose that for all i ∈ max(P) and all j ∈ [n]\ max(P) we have j < i .Denote by Q = ([l],≤) the restriction of the partial order ≤ on [l]. Then the poset weightof g = (g1, . . . , gn) ∈ G is given by wtP(g) = wtQ(g1, . . . , gl) if (gl+1, . . . , gn) = 0 andwtP(g) = l + wtH(gl+1, . . . , gn) otherwise, where wtH is the Hamming weight. Similarly,wtP(χ) = wtH(χl+1, . . . , χn) if (χ1, . . . , χl) = ε and wtP(χ) = n − l + wtQ(χ1, . . . , χl)

otherwise. Hence restricting the elements of the blocks Pm, Pm of the partitions PP, PPto the index set [l] leads to (Pm)|[l] = Qm and (Pm+n−l)|[l] = Qm, m = 0, . . . , l, whereQm, Qm are the blocks of PQ and PQ.

In the following remark we fix convenient index notation for hierarchical posets and forthe factors of the group G. It leads to a simple formula for the poset weight.

Remark 5.3 Let P be the hierarchical poset H(n; n1, . . . , nt ). Note that∑t

i=1 ni = n. Wewrite the underlying set [n] as N := {(i, j) | i = 1, . . . , t, j = 1, . . . , ni } such thatthe partial order simply reads as (i, j) < (i ′, j ′) ⇐⇒ i < i ′ (where i < i ′ refers to thenatural order on N). Consequently, for any set S = {(i1, j1), . . . , (ir , jr )} ⊆ N , wherei1 ≤ . . . ≤ is−1 < is = is+1 = · · · = ir , the ideal generated by S is 〈S] = {(i, j) ∈ N | i <ir } ∪ {(is, js), (is+1, js+1), . . . , (ir , jr )}. Accordingly, we index the factors of the group Gas Gi, j , i = 1, . . . , t, j = 1, . . . , ni . Thus,

G =t

∏

i=1

Gi , where Gi =ni∏

j=1

Gi, j , (5.2)

and g ∈ G has the form g = (g1, . . . , gt ), where gi = (gi,1, . . . , gi,ni ) ∈ Gi . Denoting bywtH the Hamming weight on each of the groups Gi and Gi , the poset weight on G and thedual poset weight on G = G1 × · · · × Gt are given by

wtP(g) =s−1∑

i=1ni + wtH(gs), where s = max{i | gi = 0}, (5.3)

wtP(χ) =t

∑

i=s′+1ni + wtH(χs′), where s′ = min{i | χi = ε}. (5.4)

For codes in Fn (thus Gi = (F,+) for all i), H. Kim and Oh [25] studied the question as to

which posets give rise to a MacWilliams identity for the induced wtP-enumerator of a codeand the wtP-enumerator of its dual. They proved that this is the case if and only if the poset ishierarchical. In an independent paper [24] D. Kim showed a similar result by making use ofthe transitivity of the isometry group on the spheres. Pinheiro and Firer [38] generalized theresult of [25] to block poset structures. We will recover these results and generalize them tocodes over groups or rings. It is pointed out that the proofs in the above-mentioned literatureare tailored to codes over fields and do not carry over to additive codes. We proceed byrelating the induced poset weight partitions to reflexivity in the sense of Definition 2.1. Hereis the first part of our result.

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Theorem 5.4 Fix a poset P = ([n],≤). Let PP and PP be the induced partitions on G = G1×· · · × Gn and G = G1 × · · · × Gn, respectively. Suppose PP = PP. Then P is a hierarchicalposet, and if P = H(n; n1, . . . , nt ) and G is as in (5.2), then |Gi,1| = · · · = |Gi,ni | forall i = 1, . . . , t .

Note that the assumption PP = PP along with |PP| = |PP| = n + 1 implies that PP isreflexive; see Theorem 2.4. But one should be aware that Theorem 5.4 does not state thatreflexivity of the poset partition PP implies that P is a hierarchical poset. We strongly believethat this is true as well, but unfortunately do not have a proof. To be more specific, supportedby many examples we believe that PP ≤ PP for any poset P. This would imply the aboveconjecture.

Proof of Theorem 5.4 Let PP = P0 | P1 | . . . | Pn and PP = P0 | P1 | . . . | Pn , where P0 ={0} and P0 = {ε}. Suppose first that P is not hierarchical. Then P is not hierarchical either.We use <P and 〈 ]P for the dual partial order and the ideals in the dual poset. Without lossof generality let min(P) = {1, . . . , l} for some l ∈ {1, . . . , n}. We may assume that thereexists j in [n]\ min(P) and i ∈ min(P) such that i <P j . (Otherwise we may disregardmin(P) and proceed with the non-hierarchical poset on {l + 1, . . . , n} since by Remark 5.2the resulting partition sets are restrictions of blocks of PP and PP.) Let now j be a minimalelement in P with the above specified property. Then 〈 j]P ⊆ (min(P) ∪ { j})\{i}, and thus|〈 j]P| =: m ≤ l. Choose χ ′ ∈ G such that supp(χ ′) ⊆ min(P) and |supp(χ ′)| = m.Moreover, let χ ′′ = (ε, . . . , ε, χ ′′

j , ε, . . . , ε) ∈ G be such that its j-th entry χ ′′j is not the

principal character of G j . Then χ ′, χ ′′ ∈ Pm . We show that χ ′ ∼PPχ ′′ and thus PP = PP.

The block Pn of PP is evidently of the form (see also [25, Lemma 2.1])

Pn = {g ∈ G |max(P) ⊆ supp(g)} = {(g1, . . . , gn) |gi = 0 for i ≤ l}.

Let |Gi | = qi . Recall from (2.1) that 〈χ, g〉 = ∏ni=1 〈χi , gi 〉 for all (χ, g) ∈ G ×G. Using

the specific form of χ and χ ′′ and the orthogonality relations (2.2) we compute

∑

g∈Pn

〈χ ′′, g〉 =l

∏

i=1

(qi − 1)n

∏

i=l+1i = j

qi

∑

g j ∈G j

〈χ ′′j , g j 〉 and

∑

g∈Pn

〈χ ′, g〉 =n

∏

i=l+1

qi

⎛

⎝

l∏

i=1

∑

gi ∈Gi \{0}〈χ ′

i , gi 〉⎞

⎠ .

Again by (2.2) the first sum is zero, whereas the second one is not. This shows thatχ ′ ∼

PPχ ′′, contradicting the assumption PP = PP. Hence P is hierarchical.

Let now P = H(n; n1, . . . , nt ) be as in Remark 5.3 and let G = G1×· · ·×Gt be as in (5.2).Assume |Gs,1| = |Gs,2| for some s. For a = 1, 2 put χa := (ε, . . . , ε, χ

(a)s , ε, . . . , ε) ∈

G1 ×· · ·× Gt , where χ(1)s := (χs,1, ε, . . . , ε), χ(2)s := (ε, χs,2, ε, . . . , ε) ∈ Gs and χs,1, χs,2

are non-principal characters. Due to (5.4) the characters χ1, χ2 have the same dual posetweight, and thus χ1∼PP

χ2.

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Put m = ∑si=1 ni . Then Pm = {(g1, . . . , gs, 0, . . . , 0) | gi ∈ Gi , wtH(gs) = ns},

see (5.3). Using S := ∏s−1i=1 |Gi | and (2.2) we compute

∑

g∈Pm

〈χ1, g〉 = S∑

gs ∈GswtH(gs )=ns

〈χ(1)s , gs〉

= Sns∏

j=2

(|Gs, j | − 1)∑

gs, j ∈Gs, j \{0}〈χs,1, gs,1〉 = −S

ns∏

j=2

(|Gs, j | − 1).

In the same way we obtain∑

g∈Pm〈χ2, g〉 = −S(|Gs,1| − 1)

∏nsj=3(|Gs, j | − 1). Now

|Gs,1| = |Gs,2| implies that∑

g∈Pm〈χ1, g〉 = ∑

g∈Pm〈χ2, g〉, and thus χ1 ∼

PPχ2. Again,

this contradicts our assumption. Thus, |Gs,1| = · · · = |Gs,ns | for all s, and this concludesthe proof. ��

The converse of the previous result is true as well.

Theorem 5.5 Let P = H(n; n1, . . . , nt ) and G be as in (5.2), where |Gi,1| = · · · = |Gi,ni |for all i . Then PP = PP for the partitions PP and PP on G and G. As a consequence, PP isreflexive.

Proof First of all, the last statement follows from |PP| = |PP| = n + 1 with Theorem 2.4.Let PP = P0 | P1 | . . . | Pn and PP = P0 | P1 | . . . | Pn . Define n0 := 0 and

Ns :=s−1∑

i=0

ni for s = 1, . . . , t + 1 and N0 := 0.

For each �,m = 1, . . . , n there exist unique indices sm ∈ {1, . . . , t} and r� ∈ {0, . . . , t−1}such that

m = Nsm + μm, where 1 ≤ μm ≤ nsm and

� = n − Nt−r�+1 + λ�, where 1 ≤ λ� ≤ nt−r� . (5.5)

Set s0 = μ0 = 0 and r0 = λ0 = 0. Then (5.3) and (5.4) show that

Pm = {(g1, . . . , gsm−1, gsm , 0, . . . , 0) ∈ G | gi ∈ Gi for i ≤ sm

and wtH(gsm ) = μm}, (5.6)

P� = {(ε, . . . , ε, χt−r� , χt−r�+1, . . . , χt ) ∈ G | χi ∈ Gi for i ≥ t − r�

and wtH(χt−r� ) = λ�}. (5.7)

Let χ ∈ G, say χ ∈ P�. We show that for each m, the sum∑

g∈Pm〈χ, g〉 does not depend

on the choice of χ , but just on the index �. It suffices to consider m > 0 and � > 0. For easeof notation write s := sm, μ := μm, r := r�, and λ := λ�. Furthermore, put |Gi, j | = qi forall j = 1, . . . , ni . Hence |Gi | = qni

i .

(1) Let s < t − r . Then∑

g∈Pm〈χ, g〉 = |Pm | = ∏s−1

i=1 qnii (qs − 1)μ

(nsμ

)

.

(2) For s = t − r we compute∑

g∈Pm〈χ, g〉 = ∏s−1

i=1 qnii

∑

gs ∈GswtH(gs )=μ

〈χs, gs〉 =∏s−1

i=1 qnii K (ns ,qs )

μ (λ), where the latter is the classical Krawtchouk coefficient from (2.5).

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(3) Finally, let s > t − r . Then∑

g∈Pm〈χ, g〉 = ∑

(g1,...,gs−1)

∑

gs∈GswtH(gs )=μ

〈χ, g〉, where the

first summation is over all (g1, . . . , gs−1) ∈ G1 × · · · × Gs−1. This equals

⎛

⎝

∑

(g1,...,gs−1)

〈(χ1, . . . , χs−1), (g1, . . . , gs−1)〉⎞

⎠

⎛

⎜

⎜

⎝

∑

gs ∈GswtH(gs )=μ

〈χs, gs〉

⎞

⎟

⎟

⎠

= 0,

where the last identity follows because the first factor is zero due to (2.2) and χt−r = ε.In all cases

∑

g∈Pm〈χ, g〉 depends only on the index � of the block P� containing χ .

Therefore PP ≤ PP and |PP| ≥ |PP| ≥ |PP| = |PP|. Thus PP = PP, which concludes theproof. ��

In the proof we also computed the Krawtchouk coefficients.

Corollary 5.6 Let P = H(n; n1, . . . , nt ) and G be as in (5.2). Assume |Gi, j | = qi for all i, j .With the notation as in (5.5) – (5.7), the Kratwchouk matrix of (PP,PP) is K ∈ C

(n+1)×(n+1),where

K�,m =

⎧

⎪

⎨

⎪

⎩

∏sm−1i=1 qni

i (qsm − 1)μm(nsmμm

)

, if sm < t − r�,0, if sm > t − r�,

∏sm−1i=1 qni

i K(nsm ,qsm )μm (λ�), if sm = t − r�.

(5.8)

Thus by Theorem 2.7, any code C ≤ G satisfies

(B0, . . . , Bn) = |C|−1(A0, . . . , An)K , (5.9)

where Am = |C ∩ Pm | and Bm = |C⊥ ∩ Pm | for all m.

These results cover several cases in the literature. Let G = Fn , which we identify with

its character module with the aid of a generating character χ as in (4.3). Note that for anyposet P the partition PP is invariant under multiplication by nonzero elements from F, andthus the dual partition does not depend on the choice of the character; see the discussion afterDefinition 4.1.

If G = Fnq (thus Gi = Fq for all i), the identity (5.9) with the Krawtchouk coefficients as

in (5.8) (where qi = q for all i) corresponds to the identity presented by Kim and Oh [25,Theorem 4.4]. Specializing even further to the hierarchical poset P = H(n; n) we simplyobtain the classical Krawtchouk coefficients and the MacWilliams identity for the Hammingweight distribution. On the other hand, specializing to the poset P = H(n; 1, . . . , 1) andusing 1 < · · · < n, we obtain the MacWilliams identity for the Rosenbloom–Tsfasmanweight, which coincides with [9, Theorem 3.1] by Dougherty and Skriganov. Pinheiro andFirer [38] generalized the results of [25] to poset block structures. In our terminology this isthe case where Gi = F

ki for all i , thus G = Fk1 × · · · × F

kn , and our results agree with thosein [38, Theorems 1, 2].

We close this section by extending the above result to matrix codes over finite Frobeniusrings endowed with a poset metric. This will come as an immediate consequence of theMacWilliams identities for the product partition and the symmetrized partition derived inSect. 3.

Let R be a finite Frobenius ring. The free module R(Rs×n) of all (s × n)-matrices over Rmay be identified with R(Rsn) by mapping M ∈ Rs×n to the vector (M1, . . . ,Ms) ∈ Rsn ,where Mi is the i-th row of M . The standard inner product v · w on Rsn thus reads as

123


Tr(M NT) on Rs×n , where Tr denotes the trace. As in Sect. 4 we identify R R with R R via afixed generating character χ .

Let P = ([n],≤) be a hierarchical poset on [n]. The product partition enumerator of acode C ⊆ R(Rs×n) is given by

PE(PP)s ,C =∑

M∈C

s∏

i=1

Xi,wtP(Mi ) ∈ C[Xi, j | i = 1, . . . , s, j = 0, . . . , n].

The coefficient of∏s

i=1 Xi,mi is thus the cardinality of all matrices in C whose i-th row

has poset weight mi for i = 1, . . . , s. By Theorem 3.3(a), (PP)s = (PP)s = (PP)

s . HenceTheorem 3.4 yields

PE(PP)s ,C⊥(Xi , i = 1, . . . , s) = |C|−1PE(PP)s ,C(K Xi , i = 1, . . . , s), (5.10)

where Xi = (Xi,0, . . . , Xi,n)Tand K is the Krawtchouk matrix of (PP, PP). In the same way,

Theorem 3.5 yields a MacWilliams identity for the symmetrized partition (PP)ssym on Rs×n .

If P is the chain 1 < · · · < n and R = Fq is a field, hence wtP is the Rosenbloom–Tsfasmanweight, the MacWilliams identities for the enumerators PE(PP)s ,C and PE(PP)ssym

with thecorresponding Krawtchouk coefficients were derived by Dougherty and Skriganov in [9,Theorems 3.1 and 3.2]. Therein, instead of character theory direct technical computationstailored to the specific situation of this weight were used. Another direct computational proofwas presented by Trinker [43].2

We conclude by mentioning that the cumulative Rosenbloom–Tsfasman weight, defined aswtP(M) := ∑s

i=1 wtP(Mi ) and introduced in [39], does not satisfy a MacWilliams identity.An example was presented in [9].

Acknowledgments The author was partially supported by the National Science Foundation grants #DMS-0908379 and #DMS-1210061. I would like to thank Marcus Greferath and Navin Kashyap for very inspiringsuggestions concerning this research project. A major part of the final write-up took place during a researchstay at the University of Zürich, and I am grateful to Joachim Rosenthal and his research group for the generoushospitality. I also would like to thank the anonymous reviewers for very helpful suggestions, in particular withrespect to the proof of Theorem 3.3(b).

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