AbstractThe number of words of weight w in the product code of linear codes with minimum distances dr and dc is expressed in the number of low weight words of the constituent codes, provided that w < drdc + max(dr, dc). By examples it is shown that, in general, the full weight enumerator of a product code is not completely determined by the weight enumerator of its constituent codes

Baggen, C.P.M.J.

Tolhuizen, L.M.G.M.

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On the weight enumerator of product codes 

Discrete Mathematics 106/107 (1992) 483-488 
North-Holland 
483 
On the weight 
product codes 
enumerator of 
L.M.G.M. Tolhuizen and C.P.M.J. Baggen 
Philips Research Laboratories, P. 0. Box 80. OlW, 5600 JA Eindhoven, Netherlands 
Received 18 December 1991 
Revised 20 January 1992 
Abstract 
Tolhuizen, L.M.G.M. and C.P.M.J. Baggen, On the weight enumerator of product codes, 
Discrete Mathematics 106/107 (1992) 483-488. 
The number of words of weight w in the product code of linear codes with minimum distances 
d, and d, is expressed in the number of low weight words of the constituent codes, provided 
that w <d,d, + max(d,, d<). By examples it is shown that, in general, the full weight 
enumerator of a product code is not completely determined by the weight enumerator of its 
constitutent codes. 
1. Introduction 
Nowadays, error correcting codes have found widespread application in all 
kinds of industrial products. Only a few decades ago, the theory has been 
founded. It almost immediately found applications in deep space communications 
and military and professional areas. Recently, the advance of digital integrated 
circuitry led to the application of coding in all kinds of consumer products. 
Especially in storage-type applications, it turned out to be beneficial to use 
algebraic block codes. For instance in optical recording, the extremely high 
information densities (=l bit/pm2) imply a raw error rate that is unacceptable for 
most digital storage applications. The use of coding, in particular Reed-Solomon 
codes, made it into a success, i.e., high storage density, combined with high 
reliability. 
The best known example today probably is the compact disc [3,7], which has 
been introduced in the beginning of the eighties. Since then, we have seen the 
introduction of the digital audio tape recorder [2] and, recently, the digital 
compact cassette (DCC) [4]. All these systems use a form of error correcting 
Correspondence to: L.M.G.M. Tolhuizen, Philips Nat. Lab. (Wy B), P.O. Box 80.000, 5600 JA 
Eindhoven, Netherlands. 
0012-365X/92/$05.00 0 1992- Elsevier Science Publishers B.V. All rights reserved 
484 L. M. G. M. Tolhuizen, C. P. M.J. Baggen 
codes based on Reed-Solomon codes, mainly because of their MDS property 
combined with the existence of efficient decoding algorithms [I, 51. Another 
similarity between these systems is that they all use a concept of what might be 
called ‘co-operating codes’, namely Cross Interleaved Reed-Solomon codes 
(CIRC) for CD and product codes of Reed-Solomon codes for DAT and DCC. 
The popularity of cooperating codes is due to their good performance, including 
protection against burst errors, combined with a reasonable decoder complexity 
[2,3,7,81. 
For performance evaluations and design of error correcting systems, the error 
event rates (both uncorrectable errors and miscorrections) must be calculated. 
Especially the behaviour of a code and a chosen decoding algorithm in the case 
that error patterns exceed the guaranteed correction capabilities of the combina- 
tion is of interest. It turns out that the weight enumerator of a code is of prime 
importance in these calculations [l, Chapter 141. 
In this article, we restrict ourselves to a simple special case of cooperating 
codes, namely to product codes, defined as follows. Let C, and C, be two block 
codes. The product code C,, = C, x C, consists of all matrices for which all 
columns are in the ‘column code’ C, and all rows are in the ‘row code’ C,. 
Throughout this paper, we will assume that C, and C, are an [n,, k,, d,] code and 
an [n,, k,, d,] code, respectively, both over GF(q). Under the assumptions, C,, is 
an [n,n,, k,k,, d,d,] code [5, Chapter 181, [l, Chapter lo]. 
In [8], several aspects of the (theoretical) error correcting capabilities of C,, are 
investigated. It includes an expression for the number of words of weight w of C,, 
in the number of low weight words of C, and C,, provided that 
w < d,d, + max(d,, d,). 
For the sake of completeness, this expression and its derivation are given in the 
second section. 
It is natural to wonder if it is possible to derive the number of words of weight 
w in C,, from the weight enumerators of C, and C, if w does not satisfy the above 
inequality. The examples in the third section provide the negative answer. 
2. Low weight words of a product code 
In this section, the low weight words of C, are described and counted [8]. 
We define the support of a word as the set of its nonzero coordinates. The 
following lemma is well known, but rarely stated explicitly. 
Lemma 1. Let d be the minimum distance of a given linear code. If x and y are 
two codewords of weight d with the same support, then x is a multiple of y. 
Proof. Let i be such that yj # 0 and define & := xi/y,. Then c := cuy --x is a 
codeword with zeros outside the support of x and at least one zero inside the 
On the weight enumerator of product codes 485 
support of x, viz. on position i. Consequently, the weight of c is less than d and so 
c=o. 0 
Theorem 2. Let X be a nonzero word in C, of weight less than d,d, + d,. There 
exists a word c, E C, of weight d, and a word c, E C, such that Xl,j = (c~)~(c,)~ for all 
i and j. 
Proof. As the weight of any nonzero row of X is at least d,, X has at least d, 
nonzero columns. If X would have at least d, + 1 nonzero columns, then X would 
have weight at least d,(d, + l), as each nonzero column of X has weight at least 
d,. As the weight of X is less than d,d, + d,, X has exactly d, nonzero columns. 
So any two nonzero rows of X have weight d, and the same support. From 
Lemma 1 it follows that two such rows are multiples of each other. Consequently, 
there exists a word c, E C, and scalars a(l), a(2), . . . , a(n,) such that the ith row 
of X equals a(i)c,. The jth column of X equals (c,)j(a(l), a(2), . . . , a(n,))T and is 
a column code word. As there is some j such that (c,)j #O, 
(a(l), a(2), . . . , a(n,))T is in fact a column code word. q 
Using Theorem 2, we can completely classify the codewords of weight less than 
d,d, + max(d,, d,). In particular, we can count the number of such code words. 
This is done in the following theorem. 
Theorem 3. Let A,(w), A,(w) and A,(w) denote the number of words of weight w 
in C,, C, and C, respectively. Suppose that d, s d,. If 1 G w < d,d, + d,, then 
0 if w is not divisible by d,, 
A,(w/d,)A,(d,)/(q - 1) if w is divisible by d,. 
Proof. Let X be a nonzero code word of weight w < d,d, + d,. From Theorem 2, 
it follows that there exist a word c, E C, of weight d, and a word c, E C, such that 
X.j = (cc)i(c,)j* Th is implies that wt(X) = d, wt(c=), so w is a multiple of d,. 
Clearly, for each nonzero scalar (Y we have X;,j = (~C,.);(c.U-‘C~)j, so X may be 
‘represented’ at least (q - 1) times. More representations are not possible as all 
nonzero rows of X have the same support as c, and therefore are multiples of c, 
(as c, has weight d,, cf. Lemma 1). As there are A,(d,) words of weight d, in C, 
and A,(w/d,) words of weight w/d, in C,, the theorem follows. 0 
Corollary. If d,d, < w c d,d, + min(d,, d,), then C,, has no words of weight w. 
3. The weight enumerator of a product code is not determined by those of its 
constituent codes 
In the previous section we expressed the number of words of weight w in C, in 
the weight distribution of the constituent codes, under the condition that 
486 L. M. G. M. Tolhuizen, C.P.M.J. Baggen 
w < d,d, + max(d,, d,). In this section, we will show that this cannot be done for 
w 2 d,d,. + max(d,, d,.). We do so by giving specific examples of product codes 
with an unequal number of words of weight d,d,. + max(d,, d,.), although the 
weight enumerators of their constituent codes are equal. The first example is 
simple and deals with binary codes. 
Example 1. Let C, be the binary [6,3,2] code with generator matrix (I 1 I), 
where I denotes the 3 X 3 identity matrix, and let C2 be the binary code with 
generator matrix 
100001 
i 1 010001 001111 . 
Both codes have weight enumerator 1 + 3z2+ 3z4 + zh. Let B be the binary 
[3,2,2] even weight code. It is obvious that the weight of a word C, x B is a 
multiple of four: its columns have even weight and its first, second and third 
column equal its fourth, fifth and sixth column, respectively. So surely, C, x B 
does not have a word of weight six. However, CZ X B has six words of weight six. 
They may be described as 
(a~a;ooo(a, +a#), 
where a, and a2 are two different nonzero words of B. 
In fact, the weight enumerators of the two product codes can easily be found by 
hand; C1 x B has weight enumerator 1 + 9z4 + 27z8 + 27~‘~ and C, X B has 
weight enumerator 1 + 9z4 + 6zh + 9zx + 18~“’ + 21~‘~. 
One might wonder if something can be said on the weight enumerator of C,, if 
the constituent codes enjoy some additional properties. Our next example shows 
that even product codes whose constituent codes are MDS and isometric’ need 
not have the same weight enumerator. 
Example 2. Let m 2 4. Let (Y E GF(2”) be such that (Y #O and (Y~’ # 1. Let C, 
and C2 be the [4,2,3] codes generated by 
1011 ( 1 Olla and 
respectively. The mapping @ : (.x1, x2, x3, x4) - (XT, xz, x:, ~24) preserves Ham- 
ming distance and maps C, onto CZ; hence, Cr and C2 are isometric. 
Both C, x C, and C, x C, have 4(q - l)(q - 3) words of weight 12 with one all 
zero row. Indeed, such a word is uniquely determined by the position of its 
’ The codes C and D are called isometric if they have the same number of words and there exists a 
mapping f from V = (GF(q))” into itself that preserves Hamming distance (i.e., d(f(x), f(y)) = 
d(x, y) for all x, y E V) and maps C onto D. 
On the weight enumerator of product codes 4x7 
all-zero row, which can be chosen in four ways, and its upper nonzero row, which 
can be chosen in A,(4) = (q - l)(q - 3) ways. (Note that any [4,2,3] code over 
GF(q) has (4 - l)(q - 3) words of weight four [5, Chapter 111.) In the same way 
it is shown that both product codes under consideration have 4(q - l)(q - 3) 
words of weight twelve with one all zero column. 
The code C, X C, has, up to scalar multiples, two words of weight 12 with no 
all-zero row and no all-zero column. Those two words are 
The code C, x C2, however, has no such words. One way of showing this is 
writing down all 4! = 24 ways to place zeroes in a 4 x 4 array with one zero in 
each row and each column, starting to make a word of C, x C2 and seeing that in 
all cases it cannot be completed. We give a slightly more elegant proof. Suppose 
that C, x C, contains a word of weight 12 with no all-zero row and no all-zero 
column. We describe this word as 
M=[aTIbTIuT+bT]aT+~bT] witha,bE&. 
Let ii denote the index of the zero in the jth column of M. Then all i, are distinct 
and ai, = bj, = ai, + bj, = ail + Ly6;, = 0, so 
ai,/a,j = a(b,lbi,). (*) 
Now note that the weight 3 words of CZ are the scalar multiples of 
(0, 1, 1, a’), (1, 0, 1, l), (1, 1, 0, 1 + cu’) and (a’, 1, 1 + (Y’, 0). This implies that if 
a is a word of weight 3 in CZ with u,u,#O, then al/u2 E (1, LY’}. As cuf my2 and 
a# cr-*, (*) cannot hold for {i3, i4} = (1, 2). One easily checks that for the other 
choices for {i?, id}, (*) d oes hold neither, due to the constraints on LY. 
Remark. It is common [5] to call two codes C and D over GF(q) equivalent if 
they have the same cardinality and there exists a permutation o E S,, and non-zero 
elements A,, . . . , A, of GF(q) such that 
~(C,,_..,C”~~C[(~I~~(,), . . . , Lc,~,,) E Dl. 
Clearly, equivalent codes are isometric. It is also easy to see that A x C is 
equivalent to B x D whenever both A is equivalent to B and C is equivalent to D. 
(Remember that all codes are linear!) As equivalent codes have the same weight 
enumerator, this observation implies that the isometric codes C, and C2 from 
Example 2 are not equivalent! In [6], it is shown that two isometric linear codes 
over a prime field are necessarily equivalent. 
488 L. M. G. M. Tolhuizen, C. P. M.J. Baggen 
References 
]I] R.E. Blahut, Theory and Practice of Error Control Codes (Addison-Wesley, Reading, MA, 
1983). 
[2] K. Hayashi et al., Error correction method for R-DAT and its evaluation, Proc. ICASSP (1986) 
9-13. 
[3] H. Hoeve, J. Timmermans and L.B. Vries, Error correction and concealment in the compact disc 
system, Philips Tech. Rev. 40 (6) (1982) 166-172. 
[4] G.C.P. Lokhoff, DCC-digital compact cassette, IEEE Trans. Consumer Electr. 37 (3) (1991) 
702-706. 
[5] F.J. MacWilliams and N.J.A. Sloane, The Theory of Error-Correcting Codes (North-Holland, 
Amsterdam, 1977). 
[6] A. Montpetit, Note sur la notion d’equivalence entre deux codes lineaires, Discrete Math. 65 
(1987) 177-185. 
[7] J.B.H. Peek, Communication aspects of the compact disc digital audio system, IEEE Comm. 
Magazine 23 (2) (1985) 7-15. 
[8] L. Tolhuizen and S. Baggen, On the correcting capabilities of product codes, Proc. 10th Symp. on 
Information Theory in Benelux, (1989) 45-50. 


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