Codes and lattices in the lp metric

Sueli I. R. CostaInstitute of Mathematics

University of Campinas, Brazilsueli@ime.unicamp.br

Antonio CampelloInstitute of Mathematics

University of Campinas, Brazilcampello@ime.unicamp.br

Grasiele C. JorgeInstitute of Science and Technology

Federal University of Sao Paulo, Brazilgrasiele.jorge@unifesp.br

Joao E. StrapassonSchool of Applied Sciences

University of Campinas, Braziljoao.strapasson@fca.unicamp.br

Claudio QureshiInstitute of Mathematics

University of Campinas, Brazilcqureshi@gmail.com

Abstract—Codes and associated lattices are studied in thelp metric, particularly in the l1 (Lee) and the l∞ (maximum)distances. Discussions and results on decoding processes, classi-fication and analysis of perfect or dense codes in these metricsare presented.

Keywords—Codes and lattices, lp metric, Lee metric, perfectcodes.

I. INTRODUCTION

We consider here integer lattices in the lp norm and q-arycodes in the induced metric in Zn

q . We call this induced metric,the p-Lee metric. For p = 1 and p = 2, those are the well-known Lee and standard Euclidean metric. Codes and latticesin the Lee metric have deserved a lot of attention lately due tonew and diverse applications (see [1] and references therein).

There are not many references of lattices considered inthe lp metric in Rn for p 6= 1, 2, in the literature. Peikert [2]studies the complexity of some important computational latticeproblems such as the closest vector problem and the shortestvector problem in the lp norm for 2 ≤ p ≤ ∞. In [3], Grellet al. show optimal algorithms in these norms for the closestpoint search problem in some well-known families of lattices,such as Zn, Dn, An, E6, E7 and E8.

In the lp metric, the decoding proccess of a ConstructionA lattice can be reduced to the decoding of its generator code(Proposition 1).

Concerning the existence of perfect codes in the p-Leemetric, we could find that there is a variety of perfect codesfor p > 1. This is in contrast with the classic Lee (or 1-Lee)metric, for which it is conjectured (Golomb and Welch [4])that there are no perfect codes in Zn, unless n = 2 or R = 1.

In the case of the ∞-Lee metric, perfect codes are char-acterized in Proposition 3. We found that the ∞-Lee metriccaptures much of the essence of perfect codes in other p-Leemetrics since any perfect code in the ∞-Lee metric is alsoperfect in the p-Lee metric for large enough p, as shown inProposition 4.

Further discussions on group characterization of perfectcodes and dense codes in the lp metric are also presented here.

II. CODES AND LATTICES

We consider here a q-ary linear code C ⊆ Znq as a Zq-

submodule of Znq , q ∈ N. If q is a prime number, then C

is a vector subspace of Znq and therefore has a basis with

k ≤ n vectors. Otherwise, we can only assure the existence ofa minimal set of generators which are not necessarily linearlyindependent [5]. In the case q = pe, p prime, these codes havegenerator matrices in a standard form [6].

A lattice Λ is a discrete additive subgroup of Rn. Equiv-alently, Λ ⊆ Rn is a lattice iff there are linearly independentvectors v1, . . . ,vm ∈ Rn such that Λ is the set of allinteger linear combinations,

∑mi=1 αivi, αi ∈ Z. The set

{v1, . . . ,vm} is called a basis for Λ. A matrix M whosecolumns are these vectors is said to be a generator matrixfor Λ. The determinant of a full rank lattice Λ, (m = n), canbe defined as det Λ = |detM | and it is an invariant underbasis change. A lattice Λ is called integer if Λ ⊆ Zn.

The so-called Construction A associates to a q-ary codeC ⊆ Zn

q an integer lattice ΛA(C) via the surjective map φ[7]:

φ : Zn −→ Znq

(x1, . . . , xn)t 7−→ (x1, . . . , xn)t,(1)

where xi = xi (mod q) for i = 1, . . . , n, ΛA(C) = φ−1(C).It is straightforward to see that ΛA(C) is a lattice if, andonly if, C is a linear code. In this case, ΛA(C) is called theq-ary lattice associated to C. Any q-ary lattice has qZn asa sublattice and this property can be used as an alternativedefinition to q-ary lattices, as done in [8]. Any integer latticeΛ can be considered as a q-ary lattice since for q = det Λ,Λ ⊂ qZn.

III. THE p-LEE METRIC

Instead of the usual Hamming metric for codes and Eu-clidean metric for lattices we consider here the lp norm forΛA(C) ⊂ Zn and the induced p-Lee metric for the associatedcode C.

Let x = (x1, . . . , xn)t,y = (y1, . . . , yn)t ∈ Rn. The usuallp distance (denoted by dp) between these vectors is defined

as

dp(x,y) :=

(n∑

i=1

|xi − yi|p)1/p

if 1 ≤ p <∞ (2)

andd∞(x,y) := max{|xi − yi|; i = 1, . . . , n}. (3)

As it is well-known, those distances are related by

nd∞(x,y) ≥ d1(x,y) ≥ dp(x,y) ≥ d∞(x,y),

for 1 < p <∞.

The Lee metric for codes, introduced for codes in [9], canbe viewed as the distance in Zn

q induced by l1 (or Manhattan)metric in Zn. For two elements x, y ∈ Zq

dLee(x, y) = min {(x− y) (mod q), (y − x) (mod q)} , (4)

whereas for two vectors x,y ∈ Znq , we have

d1,Lee(x,y) =n∑

i=1

min {(xi − yi) (mod q), (yi − xi) (mod q)} .

(5)

We define the p-Lee metric through the next proposition.

Proposition 1. [10] Let 1 ≤ p <∞ and let dp be the metricin Rn given by Equation (2). For x = (x1, · · · , xn), y =(y1, · · · , yn) ∈ Rn/qZn, 0 ≤ xi, yi < q, the induced metricin Rn/qZn by the metric dp is given by

d(x,y) =

(n∑

i=1

(dLee(xi, yi))p

)1/p

, (6)

where dLee(x, y) is given by Equation (4). We will denote thisdistance as dp,Lee(x,y) = d(x,y).

For p =∞ a similar result holds, replacing (6) by

d(x,y) = max{dLee(xi, yi); i = 1, . . . , n}.

This distance is denoted as d∞,Lee(x,y).

It is worth noting that the p-Lee metric defined as aboveappears as “generalized weights” of a code in [11] and also,for p = 2, in the context of physical layer network coding in[12].

In this paper we consider p integer, 1 ≤ p < ∞, andp = ∞, and denote by dp,Lee(C) the minimum distance ofa code C ⊆ Zn

q . The closed and open balls of radius R (Ra positive real number) in Zn

q centered at x, are denoted byBp,Lee[x, R] and Bp,Lee(x, R), respectively.

We define the packing radius R of a code C ⊆ Znq

(⊆ Zn) as the greatest R such that the balls of radius R,Bp,Lee[x, R] ⊆ Zn

q (⊆ Zn), centered at the distinct points ofC are disjoint and there is at least one point of Zn

q at theboundary of Bp,Lee[x, R].

For p = 1 and p = ∞ the packing radius of a code C ⊂Znq is integer and given by the well-known expression R =b(dp,Lee(C)− 1)/2c. For 1 < p <∞, a similar expression isnot valid (Fig. 1).

Fig. 1. Packing balls for the code C = 〈(1, 5)〉 ⊆ Z213 in the 2-Lee metric

(R = 2 and d2,Lee(C) =√

13).

r

r Hmod 13L

x

-10 -5 5 10

-5

5

10

Fig. 2. Lee Decoding r = (0,−6) ∈ R2 in ΛA(C), C =⟨(1, 5)t

⟩.

IV. DECODING CONSTRUCTION A LATTICES

For q-ary lattices constructed from q-ary codes, we haveshown in [13] that decoding a q-ary code C ⊆ Zn

q consideringthe Lee metric corresponds to decoding the associated q-arylattice ΛA(C) ⊆ Rn considering the sum metric l1. In [10],we got the same kind of relation between codes in the p-Leemetric and lattices in the lp metric.

In order to simplify the notation, x is a codeword of C,whereas x, is a point of ΛA(C). Due to the isomorphismΛA(C)/qZn ' C, we will not distinguish the elements ofΛA(C)/qZn ⊆ Rn/qZn from the codewords of C.

Proposition 1. [10] Let ΛA(C) be a q-ary lattice andr = (r1, . . . , rn)t ∈ Rn. Let r ∈ Rn/qZn and c ∈ C,c = (c1, . . . , cn)t, 0 ≤ ci < q, a closest codeword to rconsidering the p-Lee metric in Rn/qZn. An element z ∈ΛA(C) which is closest to r considering the lp metric in Rn is

z = (z1, · · · , zn)t, where zi = ci +qwi and wi =

⌈ri − xiq

⌋,

for each i = 1, . . . , n.

Example 1. Consider the cyclic 13-ary code in Z213, C =⟨

(1, 5)t⟩. It has minimum Lee distance d(C) = 5 and error

correction capacity t = 2. For the received vector r = (0,−6)t

the Lee-closest codeword to r = (0, 7)t is x = (12, 8)t. Theclosest lattice point to r is z = (−1,−5)t. Fig. 2 shows thelattice ΛA(C) and its Lee Voronoi regions.

V. PERFECT CODES IN THE p-LEE METRIC

Let d be a metric in Znq , and C a code. If C has the property

that, for any x ∈ Znq there is only one codeword c ∈ C such

that d(c,x) ≤ R we say it is a perfect code with radius R.The trivial perfect codes are C = Zn

q and C = {0}.

The characterization of perfect codes is completely solvedin the case of the Hamming metric and q a prime power andit is an open problem in the Lee (1-Lee) metric, in which casethe famous Golomb-Welch [4] conjecture attempts to providean answer.

Analogously, a (linear) perfect code with radius R in Zn

is a lattice Λ ⊂ Zn such that, for each point x ∈ Zn, there isonly one c ∈ Λ satisfying d(x, c) ≤ R. These perfect codesare also called tilings of Zn.

If C is a q-ary code, the minimum distance µ =min{dp(x,0),0 6= x ∈ ΛA(C)} of ΛA(C) is related to theminimum distance dp,Lee(C) of C, µ = min{q, dp,Lee(C)}[11]. If dp,Lee(C) ≤ q, then finding perfect codes in Zn

q in thep-Lee metric is equivalent to finding tilings in Zn. Otherwise,we may have a perfect code in Zn

q that does not induce a tilingin Zn, as shown in the next example

Example 2. The binary code C ={(0, 0, 0, 0, 0, 0, 0), (1, 1, 1, 1, 1, 1, 1)} ⊆ Z7

2 is perfect inthe Lee metric. However, the associated 2-ary lattice ΛA(C)does not tile Z7.

In this paper we only will consider codes in Znq for which

dp,Lee(C) ≤ q. Under this hypothesis, we have the followingSphere-Packing bounds. If µp(n,R) = |Bp,Lee(0, R)|, then|C|µp(n,R) ≤ qn for a code in Zn

q with packing radius R.For a lattice Λ ∈ Zn (linear code), |det Λ| ≥ µp(n,R).

For p = 1 and p =∞, there are closed forms for µp(n,R),namely

µ1(n,R) =

min{n,R}∑i=0

2i(n

i

)(R

i

)(7)

andµ∞(n,R) = (2R+ 1)n. (8)

The equality in these bounds will be achieved for perfect codes.

For p = 1, the long standing Golomb-Welch conjecture,stating that there is no perfect codes in the Lee metric in Zn

for n > 2 and packing radius R > 1, is widely believed tobe true but has, up to now, only been proved in special cases[14]. Quasi-perfect, diameter perfect and dense codes have alsobeen approached in recent papers [15],[1] and [16].

Perfect 1-Lee codes in dimension two (linear and nonlin-ear) can be fully characterized. Let PL(2, R, q) be the set ofperfect codes in the 1-Lee metric correcting R-errors (packingradius R) in Z2

q .

Proposition 2. [17] Let qR = 2R2+2R+1 and ν1 = (R,R+1), ν2 = (−(R + 1), R), η1 = (1,−(2R + 1)), η2 = (0, qR)be vectors in Zq × Zq . Let DR = ν1Z + ν2Z and ν be theconjugate of ν. We have the following characterization:i- PL(2, R, q) 6= ∅, if and only if, q 6= 0 (mod qR) whereqR = 2R2 + 2R+ 1.ii- C ∈ PL(2, R, q) if and only if C = c+De or C = c+DR

for some c ∈ C. (in particular C − c is a group).iii- If C ∈ PL(2, R, q) and GC = C−c is the group associatedwith C, then: GC is cyclic if and only if q = qR. In such caseGC ' Zq with generator ν1 = (R,R+ 1) if GC = DR or ν1if GC = DR. If q = hqR with h > 1, then GC ' Zq × Zh.

Explicitly, GC = η1Z⊕ η2Z or GC = η1Z⊕ η2Z if GC = De

or GC = DR, respectively.

Fig. 3. A non-cyclic 1-Lee perfect code in Z226 generated by vectors (1, 21)

and (0, 13) (R = 2 and q = 2qR).

In some examples, we can see that it could be expecteddifferent results concerning the existence of perfect codes inthe p-Lee metric for p 6= 1. In opposition to the 1-Lee metric,there is no perfect p-Lee code in Z2 for p = 3 and R = 4(the associated packing ball of radius 4 does not tile Z2). Fig4 shows a code with p = 3, R = 4 and “packing density” inZ2 equals to 53/60.

p = 3, R = 4

Fig. 4. Lattice generated by (4, 7) and (8,−1)

A natural question to be raised is if and how certainproperties of the 1-Lee metric can be extended to the p-Leemetric. We summarize next some results which are, in part, aresearch in progress.

The following proposition is an extension of a result in [4].

Proposition 2. [10] For 1 ≤ p < ∞, there are perfect codesin Zn

q in the p-Lee metric for R = 1 and q = 2n+ 1.

Perfect codes in the case p =∞ can be fully characterizedthrough the following proposition:

Proposition 3. [10] There are non-trivial perfect codes C ⊂Znq in the ∞-Lee metric iff q = bm with b > 1 an odd integer

and m > 1 an integer.

The codes obtained in the proof of this proposition areCartesian products but, as it can bee seen in Fig. 5, there areother families of perfect codes in the ∞-Lee metric. A full

classification of such codes as in Prop. 2 is an interesting topicof investigation.

Fig. 5. The cyclic code C =⟨(1, 7)

⟩⊆ Z2

49, which is perfect in the ∞-Leemetric

The next proposition shows that for each perfect code inthe ∞-Lee metric there exists p∗ ≥ 1 such that this code isalso perfect in the p-Lee metric for all p ≥ p∗.Proposition 4. Let C ⊆ Zn

q be a perfect code in the ∞-Leemetric with packing radius R. If p > ln(n)

ln(1+ 1R )

, then C is

perfect in the p-Lee metric.

Fig. 6. A perfect code in the p-Lee metric for 2 ≤ p ≤ ∞. Packing balls forp = 2 on the left and for p = 4 on the right.

n 10 25 50 100 500 1000 5000 10000p 6 8 10 12 16 18 22 23

TABLE I. SMALLEST VALUES OF p SUCH THAT THERE ARE PERFECTCODES IN THE p-LEE METRIC IN Zn

10

As similar non-existence result of [4] for p = 1 can bederived for 1 < p <∞, as stated in the next proposition. Theproof is based on the association of perfect codes with tilingsof Rn by polyominoes, as done for example in [18].

Proposition 5. [10] Let n > 2, and 1 < p < ∞. There isRn,p such that for R > Rn,p there is no code C ∈ Zn in thep-Lee metric that reaches the Sphere-Packing bound.

Further questions to be investigated should be to detectdense codes in the cases where no perfect codes exist, as thesearch for diameter perfect or quasi-perfect codes in the p-Leemetric [1], [14], [16].

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