15.1 general properties of prime-length composite codes

Prof. Jen-Fa Huang, Fiber-Optic Communications Lab.Prof. Jen-Fa Huang, Fiber-Optic Communications Lab.National Cheng Kung University, Taiwan.National Cheng Kung University, Taiwan.

15.1 General Properties of Prime-Length Composite Codes

Let F be the binary field GF(2) and Fn the vector space of all binary n-tuples. Let Vi be an (ni, ki) binary linear code for i=1, 2, where n1 and n2 are relatively prime.

With n=n1n2, the function i, i=1, 2, defined by

i： Vi → (vi, vi, ..., vi) (15.1.1)

where vi Vi and is repeated n/ni times, is an injective linear map from Vi into Fn. Let ~Vi be the image of Vi under i. Then ~Vi is a linear subspace of Fn of dimension ki.

By combining in a bit-by-bit modulo-2 addition fashion, we get a new linear code V(2) defined by

V(2) = ~V1 + ~V2 (15.1.2)



This code has length n = n1n2 and dimension

k = k1 + k2 – dim( ) (15.1.3)

Since n1 and n2 are relatively prime, it easily follows that

1, if 1n1V1 and 1n2 V2 dim( ) = { (15.1.4) 0, otherwise,

where 1ni, for i=1, 2, is the all-ones word of length ni.

~V

1~V

2

~V

2~V

1


THEOREM 15.1:

If Vi is an (ni, ki) binary linear code for i=1, 2, and n1 and n2 are relatively prime, then the binary linear code V(2), as defined in (15.1.2), is an (n, k) code with

n = n1n2 and k = k1 + k2 - ,

where =1 if both V1 and V2 contain the all-ones word and =0 otherwise.

For case of Vi, i=1, 2, being M-sequence code, there is no all-ones codeword exists, so =0, and the dimension of V(2) is k = k1 + k2.



The above statements can be clarified by examining the example of Figure 15.1. With suitable initial states loaded on the memory cells and with the shift register feedback circuits connected like that shown in Figure 15.1(a) (i.e., with h1(X) = 1+X+X2 and h2(X) = 1+X+X3), the M-sequences codes V1 and V2 of periods n1 = 3 and n2

= 7 will respectively be generated from their shift register circuits.

The corresponding code bits of these two code sequences are modulo-2 added together during each unit of clock time and results in the prime-length composite code of period n = n1n2 = 21, as can be seen from Figure 15.1(b).



(a) ~

( )V1 x 110110110110110110110~

( )V2 x 111001011100101110010

V(2)( )x 001111101010011000100

(b)

+

Figure 15.1: An example for generating prime-length composite code.



Note that with all possible initial loadings on the above shift register code generator, the possible codewords number of V1 is 2k1 = (1+n1) = 4, and the possible codewords number of V2 is 2k2 = (1+n2) = 8. Moreover, none of these words is the all-ones word.

Hence the possible number of codewords of the prime-length composite code V(2) is 2k1+k2 = (1+n1)(1+n2) = 32.



The codewords of the prime-length composite code V(2) are distributed among the following four cyclic classes:

1). One cyclic class is the all-zeros word. This corresponds to the case of both V1(X) and V2(X) are zero codewords;

2). One cyclic class contains n1=3 cyclic codewords. This corresponds to the case of V1(X)≠0 and V2(X)=0;

3). One cyclic class contain n2=7 cyclic codewords. This corresponds to the case of V1(X)=0 and V2(X)≠0;

4). One cyclic class contains n = n1n2 = 21 cyclic codewords. This corresponds to the case of both V1(X) and V2(X) are nonzero codewords.



Theorem 15.1 can be easily extended, by a recursive

application, to the case of (ni, ki) component codes

Vi, where ni’s are pairwise relatively prime, for any

> 2.

For Vi’s be M-sequence codes, the structure of

maximal-length code tells us that none of the ~Vi+1 is

the all-ones code, hence the dimension is deduced to

k = i=1 ki (15.1.6)



Figure 15.2 shows the commonly adopted generator configuration of prime-length composite code made up of pairwise relatively prime lengths component codes.

The prime-length composite code constructed with the configuration of Figure 15.2 has an equivalent generator polynomial.

In the Theorem 15.2 below we discuss the case when, with respect to Equ. (15.1.2), the component codes Vi’s, i=1, 2, are cyclic.



Figure 15.2: Configuration of prime-length composite code generator.



Theorem 15.2:

If Vi is cyclic with the generator polynomial gi(X), i=1, 2, then V(2) is cyclic with the generator polynomial

g(2)(X) = GCD{g1(X)(1+Xn)/(1+Xn1),

g2(X)(1+Xn)/(1+Xn2)} (15.1.7)



Chapter XV. Prime-Length Composite Coding for Chapter XV. Prime-Length Composite Coding for Multi-Users CommunicationMulti-Users Communication

The recursive application of Theorem 15.2 to the case of component codes results in the following.

Corollary 15.2:

If Vi is an (ni, ki) cyclic code with the generator polynomial gi(X), i=1, 2, …, , and (ni, nj) = 1 for i≠j, then V() made up of Vi’s is cyclic with the generator polynomial

g()(X) = GCD{g1(X)(1+Xn)/(1+Xn1),

g2(X)(1+Xn)/(1+Xn2), …,

g(X)(1+Xn)/(1+Xn)} (15.1.8)

where n = n1n2…n


Concerning the weight distribution of composite code V(2) generated by the generator polynomial g(2)(X) of Equ. (15.1.7), let us consider the following.

Suppose V1(X) have its j-th coordinate be v1j, j = 0, 1, 2, …, n1-1, i.e.

V1(X) = v10 + v11X +… + v1,n1-1Xn1-1 (15.1.9)

or in vector form,

V1 = {v10, v11, …., v1,n1-1} (15.1.10)



Similarly, suppose V2(X) have its code vector of the form

V2 = {v20, v21, …., v2,n2-1} (15.1.11)

Let be the n2-tuple formed by repeating element v1j by n2 times,

= {v1j, v1j, …, v1j} (15.1.12)

and let be the n2-tuple defined by

= V2

= {v1jv20, v1jv21, …, v1jv2,n2-1} (15.1.13)

v j~v1j

v j

~v1j

~v1j



On the basis of the nature of 1 and 2, and since n1 and n2 are relatively prime, we can conclude that the composite codeword 1(V1)2(V2) is some permutation of the n-tuple and, therefore both of them have the same weight.

Now, from (15.1.13), we know that

1). If v1j = 0, the = V2 and have the weight of w2, the number of “1” in V2;

2). If v1j = 1, then = V2, the complement of V2, and have the weight of (n2 - w2), the number of “0” in V2;

3). Within n=n1n2 bits, the possible number of v1j = 0 is (n1

- w1) and the possible number of v1j = 1 is w1.

v j

v j



Combining the above conditions, we conclude that the n-tuple … has the weight of the composite codeword 1(v1)2(v2). Hence we have the following theorem.

Theorem 15.3:

With V1, V2 and V(2) as previously defined, if viVi has weight wi, i=1, 2, then the corresponding word 1(v

1)2(v2) of V(2) has weight

W(2) = w1.(n2 - w2) + w2.(n1 - w1). (15.1.14)



Consider the case of M-sequence component codes. Recall that a binary M-sequence code is an (n=2m–1, k=m) cyclic code, all of its nonzero n=2m–1 words having the same weight of (n+1)/2 = 2m-1.

If V1 and V2 are respectively (n1, k1) and (n2, k2) M-sequence codes with (n1, n2)=1, then the weight distribution of V1 is

given by N1(0) = 1,

N1{(n1+1)/2} = n1, (15.1.15a)

and that of V2 is given by

N2(0) = 1, N2{(n2+1)/2} = n2, (15.1.15b)



With the above possible weight combinations, a direct application of Theorem 15.3 will give the weight of V(2) be

(15.1.16)

Since with respect to Theorem 15.1, = 0 or k = k1+k2

for case of V1 and V2 are both M-sequence codes, therefore,

from Eqs. (15.1.15a) -- (15.1.16) and follow the proof procedures of Theorem 15.3, we obtain the following weight distribution of code V(2).

0, if w1 = 0, w2 = 0; n1(n2+1)/2 , if w1=0, w2=(n2+1)/2;W(2) = { n2(n1+1)/2, if w1=(n1+1)/2, w2=0; (n1n2-1)/2, if w1=(n1+1)/2, w2=(n2+1)/2;



Theorem 15.4:

If Vi is an (ni, ki) M-sequence code for i=1, 2, n1 and n2 b

eing relatively prime, then the composite code V(2) has the weight distribution

N(0) = 1,

N((n1n2 - 1)/2) = n1n2,

N((n1n2 + n1)/2) = n1,

N((n1n2 + n2)/2) = n2, (15.1.17 )

where N(w) is the number of words of weight w.



Generally, in the case of component codes, the weight equation of code V() is

W() = W(-1)(n – w) + w(N(-1) – W(-1)), (15.1.18)

where W(-1) and N(-1) = n1n2…n-1 are respectively the wei

ght and the length of the composite code V(-1), while w a

nd n are respectively those of the -th component code

V.

The importance of Theorem 15.3 or Equ.(15.1.18) lies in that, given weight distributions of Vi’s that of V() ca

n be computed easily. This also means, of course, that the weight distribution of the dual of V() can be determined using the famous MacWilliams identities.



Let us examine the correlation between words of prime-length composite V() for the case when all of the component codes Vi are M-sequence codes.

Recall that an (ni=2mi-1, ki=mi) binary M-sequence code Vi has ni nonzero codewords all of which have the same weight of 2mi-1 = (ni+1)/2. Also recall that the correlation between two binary n-tuples u and v is given by (u,v) = [n - 2dis(u,v)]/n = [n - 2|u v|]/⊕ n (15.2.1)where dis(u,v) is the Hamming distance between u and v, and |u⊕v| is the Hamming weight of codeword u⊕v. For a linear code V, if both u and v are codewords of V, then u⊕v is also a word of V.

15.2 Correlation Spectra of Prime-Length Composite Codes


Let S be the set of correlations in V(). For example, as we

have known, S1 = {1, -1/n1} for M-sequence code.

With reference to (15.1.18), we note that after j M-sequence component codes have been combined into V(j), j{1, 2, ..., -1}, if we add another (j+1)th M-sequence code which has a length relatively prime to those of previous j component codes, then we will get

W(j+1) = W(j)(nj+1-wj+1) + wj+1(N(j)-W(j)), (15.2.2)

where W(j) is a weight in code V(j), W(j+1) is a weight in code V(j+1), and N(j)= n1n2…nj is the length of V(j).

Surely, the length of code V(j+1) is

N(j+1) = N(j)nj+1 = n1n2…nj+1 . (15.2.3)



Since the possible weight of component code Vj+1 is

either wj+1 = 0 or wj+1 = (nj+1 +1)/2,

therefore (15.2.2) gives

W(j)nj+1, if wj+1 = 0; (15.2.4a) W(j+1) = { [N(j+1)+N(j)-2W(j)]/2, if wj+1 = (nj+1+1)/2, (15.2.4b)



Corresponding to (15.2.4a), we have

(j+1) = [N(j+1) - 2W(j+1)]/N(j+

1)

= [N(j+1) - 2W(j)nj+1]/N(j+1)

= [N(j) - 2W(j)]/N(j),

So that,

(j+1) = (j), (15.2.5a)



Corresponding to (15.2.4b), we have

(j+1) = [N(j+1) - 2W(j+1)]/N(j+1)

= [N(j+1) - N(j+1) - N(j) - 2W(j)]/N(j+1)

= -[N(j) - 2W(j)]/N(j+1)

= -[(N(j) - 2W(j))/N(j)]/nj+1.

So that,

(j+1) = -1/[nj+1(j)]. (15.2.5b)



What (15.2.5a) and (15.2.5b) imply is that if a certain correlation value belongs to Sj, then Sj+1 contains both and -/nj+1.

Recalling that

S1 = {1, -1/n1}, (15.2.6a)

and using (15.2.5a) and (15.2.5b) recursively, we see that

S2 = S1 U (–1/n2)S1

= {1, -1/n1} U (–1/n2){1, -1/n1}

= {1, -1/n1, -1/n2, 1/n1n2}, (15.2.6b)

S3 = S2 U (-1/n3)S2

={1, -1/n1,-1/n2,1/n1n2}U(-1/n3){1,-1/n1,-1/n2,1/n1n2}

={1, -1/n1,-1/n2,-1/n3,1/n1n2,1/n1n3,1/n2n3,-1/n1n2n3} (15.2.6c)

and so on. Examining S1, S2, S3, …, we conclude the following



Theorem 15.5:

If the component codes Vi are all (ni=2mi-1, ki=mi) binary M-sequence codes, w

here i=1, 2, …, and (ni, nj)=1, then the prime- length composite code V() ha

s the set S, of correlations, defined by S = 0 U 1 U 2 U … U , where 0

is the set containing the only number of 1, and j , for j 1, is the set of num

bers of the form

(-1)(j)/ni1ni2…nij.

To clarify the statement of the Theorem 15.5, we mention, as an example, that, if = 5, then, say,

4 = {(-1)(4)/n1n2n3n4, (-1)(4)/n1n2n3n5, (-1)(4)/n1n2n4n5, (-1)(4)/n1n3n4n5, (-1)(4)/n2n3n4n5}.



Figures 15.3 and 15.4 give examples of the possible correlation values and their corresponding power spectra for the prime-length composite codes V(2) and V(3), respectively. In Figure 15.3, we use (n1=3, k1=2) and (n2=7, k2=3) M-sequen

ce codes as V(2)’s component codes, while in Figure 15.4, (n1=3, k1=2), (n2=7, k

2=3), and (n3=31, k3=5) M-sequence codes are used as V(3)’s component codes.

In either case, note that (ni, nj)=1 for i j. By cyclic shifting a typical codewor

d V(2)(X) V(2) (V(3)(X) V(3)) and counting the Hamming distance between Xl

V(2)(X) (XlV(3)(X)) and V(2)(X) (V(3)(X)) for l=0, 1, 2, …, n-1, where n=n1n2 (n=

n1n2n3) is the code length of V(2) (V(3)), then through the operation of Equ. (15.2.

1) we obtain the typical autocorrelation function of Figure 15.3(a) (Figure 15.4(a)), which obviously coincides with what we had derived in Theorem 15.5.



Corresponding to the autocorrelation function of Figures 15.3(a) and 15.4(a); the power spectra of Figures 15.3(b) and 15.4(b) are obtained from the well-known Fourier Transform relationship between correlation function and power spectrum. Alternatively, since the discussed autocorrelation function can be represented as the sum of several partial functions all of the same form but with different periods and amplitudes, the power spectrum can be approximated as the sum of several partial spectra, each corresponds to a certain partial correlation function. In any event, the resulted power spectrum is line spectrum which bears much resemblances in envelopes with that of M-sequence code. However, as we can see, the amplitudes are actually “lumpy”. This is because the based correlation functions having minor peaks within out-of-phase durations.



From Theorem 15.5 together with Figures 15.3(a) and 15.4(a) it is clear that, among the correlations 1, || 1/n1, if we assume, without losing any generality, that

n1 n2 … n. Thus, when the component codes are all

M-sequence codes, the magnitude of the correlation in the prime-length composite code V() can be kept below as small a value as desired by making the length of the shortest component code long enough.



Having derived the possible correlation values of prime-length composite codes, we now proceed to discuss the question as to the number of positions by which a codeword has to be shifted so that the correlation between the codeword and its cyclic shift is any specific value from the set of values given in Theorem 15.5.



Let VA and VB be two binary cyclic codes, VA being (nA, kA) co

de and VB being (nB, kB) code, where (nA, nB) = 1. Let VC be the

composite code, as defined in (15.1.2), with VA and VB as the c

omponent codes, so that a codeword VC(X) of VC is given by

VC(X) = VA(X) [(1+XnAnB)/(1+XnA)]

+VB(X) [(1+XnAnB)/(1+XnB)] (15.2.7)

where VA(X) belongs to VA and VB(X) belongs to VB.



Now let us consider

F(X) = VC(X) + XVC(X) mod (1+XnAnB). (15.2.8)

If = MnA, then from (15.2.7) and (15.2.8), we have

F(X) = VB(X)[(1+XnAnB)/(1+XnB)](1+XMnA) mod (1+XnAnB), (15.2.9)

which can be rewritten as

F(X) = (X) + XMnA (X) mod (1+XnAnB), (15.2.10)

where (X) is the (nAnB)-tuple obtained by repeating VB(X) nA times, i.

e., (X) = VB(X){(1+XnAnB)/(1+XnB)}. But shifting (X) cyclically by

MnA positions is the same as shifting VB(X) cyclically by B positions,

where B = MnA modulo nB.



we see that

(VB(X) + XBVB(X) mod (1+XnB)) = (nB-2)/nB, (15.2.13)

and

(VC(X) + XMnAVC(X) mod (1+XnAnB))

= (nAnB-2nA)/nAnB, (15.2.14)

Comparing (15.2.13) and (15.2.14) we see that they are equal.



Theorem 15.6 :

Suppose VA and VB are two binary cyclic codes with the re

spective lengths nA and nB such that (nA, nB) = 1. Then ever

y correlation occurring in VA (VB), for a cyclic shift of A

(B) positions, occurs also in the composite code VC for a c

yclic shift of MnB (MnA) positions, where M and A (B) ar

e related through A = MnB modulo nA (B = MnA modulo

nB).



A composite code V() with component codes V1, V2, …, V

can be treated as a composite code VC with component code

s VA and VB, where VA is a composite code with component

codes, say, Vi1, Vi2, …, Viu and VB is a composite code with c

omponent codes Vi(u+1), Vi(u+2), …, Vi[u+(-u)], where Vj {V1, V

2, …, V}. Using this fact with reference to Theorems 15.5 a

nd 15.6 in recursive fashion, we obtain the following:



Theorem 15.7:

With reference to Theorem 15.5, the correlation (-1)a/ni1ni2…n

ia occurs for a cyclic shift of bn/ni1ni2…nia positions, where b i

s a positive integer and not a multiple of ni1, ni2, …, nia, a is a p

ositive integer 1 and -1, n=n1n2…n.

With respect to Theorem 15.7, we note that for other cyclic shifts we have the correlation of (-1)/n.



To clarify the statement of Theorem 15.7, suppose = 4. Then according to Theorem 15.5, V(4) has the set S4 of correlations given by

S4 = {1, -1/n1, -1/n2, -1/n3, -1/n4, 1/n1n2, 1/n1n3, 1/n1n4, 1/n2n3, 1/n2n4, 1/n3n4,

-1/n1n2n3, -1/n1n2n4, -1/n1n3n4, -1/n2n3n4, 1/n1n2n3n4}.

If is the number of positions by which the codeword is cyclically shifted, then, according to Theorem 15.7, the correlation of 1 occurs for = 0, -1/n

i for = bn1n2n3n4/ni, 1/ninj for = bn1n2n3n4/ninj, -1/ninjnk for = bn1n2n3n4/ni

njnk, and 1/n1n2n3n4 elsewhere. We note here that b is not a multiple of any of

the numbers in the denominator for a given .


15.1 general properties of prime-length composite codes

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