A generalization of the BCH bound for cyclic codes, including the Hartmann-Tzeng bound

JOURNAL

Series A 33,229-232

OF COMBINATORIALTHEORY,

A Generalization Cyclic Codes, Including

(1982)

of the BCH Bound for the Hartmann-Tzeng Bound C. Roos

Department

of Mathematics 2628BL

and Informatics, De@ University Delfr, The Netherlands

Communicated

by the Managing

of Technology,

Editors

Received December 14. 1981

In this paper it is shown that both the BCH bound and the Hartmann-Tzeng bound for the minimum distance of a cyclic code can be obtained quite easily as consequences of an elementary result concerning the defining set of its zeros.

1. INTRODUCTION Let C be any cyclic code of length n over the field GF(q), and let m be the multiplicative order of q modulo n. If N is a set of nth roots of unity in GF(q”) such that c(x) E c 0 va E N: c(a) = 0, then N will be called a defining set of zeros for C, and we shall write c= c,. Let a be a primitive nth root of unity and b any nonnegative integer. If N contains the consecutive powers ab+ilcl, for O 2 and (n, ci) = 1, then the minimum distance d of C is a least 6. This result is known as the BCH bound for cyclic codes. See, e.g., [ 1, 3,4]. Hartmann and Tzeng observed that there exist many cyclic codes whose defining sets of zeros contain more than one set of consecutive roots, and they succeeded in improving the BCH bound for such codes [2]. In this paper we shall show (in Section 3) that both the BCH bound and the Hartmann-Tzeng bound (and also the obvious generalizations of the Hartmann-Tzeng bound) for cyclic codes are immediate consequences of our main theorem (Theorem 1, Section 2), which is an elementary result with an easy proof, 229 0097-3 165/82/050229-04$02.00/0 Copyright @Z 1982 by Academic Press, Inc. All rights of reproduction in any form reserved.

230

C. ROOS 2. THE MAIN RESULT

We start with a lemma. Let K be an arbitrary field. Then one has LEMMA 1. Let A = [a,, a*,..., a,,] be a matrix over K such that no k - 1 columns of A are linearly dependent, and let x = (x, , x2 ,..., x,) E K” have the property that no k of the elementsxi are mutually equal. Then, in the matrix A, =

al [ vl

a2

.+-

a,

x2a2

-..

x,a,

I

any k columns are linearly independent. Proof: Suppose that A’ contains k columns which are linearly dependent. Without loss of generality we may assume that these are the first k columns. Then there will exist elements Ai, A2,..., I, E K (not all zero) such that k

4 liai i=l

= T’ Aixiai

= 0.

i=l

Hence, k-l 1 i=

Ai(Xi

-

Xk)

8i

=

0.

I

Since any k - 1 columns of A are linearly independent this implies that Iz,(x, - xk) = 0 for 1 < i & k - 1. But li # 0, however, for 1 Q i < k; this also follows from the fact that no k - 1 columns of A are linearly dependent. Hence we obtain x, = xZ = . . I = xk, which contradicts our hypothesis. I Now let N be a nonempty set of nth roots of unity in GF(qm). If N= {al,a2,..., a,}, then the t x n matrix over GF(qm) whose ith row is defined as (ai, af ,..., al = 1)

will be denoted as HN. Clearly, HN is a parity check matrix for C,,,. The code over GF(q”) with HN as parity check matrix will be denoted as C,*. It contains C, as a (subfield) subcode. Therefore it is obvious that dN > d,$, where dN and d,$ denote the minimum distance of C, and C,$ respectively. For each divisor i of n we define the cyclic code c,, as follows: Put N’ := {a’ 1a E N} and n = in’. Then, if i > 1, it is clear that the cyclic code C,,,, is degenerate, because its check polynomial divides x”’ - 1; it consists of the i-fold repetitions of the words in the cyclic code of length n’ having N’ as defining set of zeros. The latter code will be denoted as CN,, and the corresponding code over GF(qm) as t?jji.

231

BCH BOUND

THEOREM 1. Let N be a nonempty set of nth roots of unity in GF(qm) such that d$ < n, and let /3 be an nth root of unity of order i. Then d,&* > d$ , and equality will occur if and only if @ has minimum distance d,*. ProojI The inequality d,&, > d*N is an obvious consequence of the inclusion N c NV PN. Equality occurs if and only if the code C,&,, contains a word of weight d,$, i.e., if and only if the matrix HNVbN contains d,* columns which are linearly dependent over the field K := GF(q”‘), Therefore, let us assume that C,&BN contains a word c = (c, , c2 ,...,c,) of weight d$. Without loss of generality we may assume that c, # 0. Now Lemma 1 implies that ck can be nonzero only if /Ik = p” = 1. Hence, ck # 0 implies that i divides k. So c$ has minimum distance d,$ in this case. Conversely, if c,$ has minimum distance d$, then there exists a word c = (c, ) c, )..., c,) E C,* such that ck # 0 holds only if i / k. Hence /3k = 1 if ck # 0, and consequently, one also has c E C$,,,. Hence d,&Bh. = d,* . 1 EXAMPLE 1. Take n = 15, q = 2, and N = {a, a2 }, where a is a primitive element in GF(24). Recall that the decomposition of x1’ + 1 in irreducible factors is as follows:

x(x4+x3+

l)(x4+x3+xZ+x+

1).

Suppose that a4 + a + 1 = 0, p = a’ for somej, o@) = i and N’ = N U /3N. If o@) = 15, then N’ = { I), hence CNi is the zero code, whence d$, > dz = 3. If The minimal polynomial of as is OGB)= 5, then N’= (a’, a*‘}. m,(x) =x2 +x + 1. This is the generator polynomial of @, which has length 3. Hence c,$ has minimum distance 3. So Theorem 1 yields that ds, =d,$ is the case. If o(J)= 3, then N’= {a’, a”}. Since m3(x) = x4 +x3 +x2 +x + 1, the code c$, which has length 5, has minimum distance 5. Therefore dz, > d,* . COROLLARY 1. If oCp) > n/d,*, then dzUDN > d,$ . ProoJ Let o@)=i length n’ its minimum result. 1

and n=in’. Then n’ < dz. Since the code c$i has distance cannot be d$. Hence Theorem 1 implies the

3. APPLICATIONS COROLLARY 2 (BCH bound). Let N = {ab+ilcl [ 0 < i, ( 6 - 21, where (n, cl) = 1 and 6 > 2. Then d$ > 6.

232

C.ROOS

ProoJ: Use induction on 6. The case 6 = 2 is obvious. Now let 6 > 2. Put P=aC 1. Since (n, c,) = 1 we have o(J) = n. Assuming that ds > 6 (induction hypothesis) we may apply Corollary 1. This yields that d&N > 6 + 1. Now observe that N U PN = {a btilcl ) 0 < i, < 6 - 1). Hence we are done. 1 COROLLARY 3 (Hartmann-Tzeng O 6 + s.

bound, where

generalized). 622,

(n,c,)=

Let

N =

1 and

Proof. Use induction on s. The case s = 0 corresponds to the situation in Corollary 1. Therefore, let us assume that s > 0 and that dz > 6 + s for the set N described in the corollary. Note that p = a’* has order n/(n, c,), which exceeds n/(6 + s) since (n, cZ) < 6. Hence Corollary 1 applies. It yields that d*NunC2N>8+~+ 1. Since NUa’*N= (abti1c1+i*c2~O 6.

It is obvious from the above proofs that Corollary 1 enables us to generalize Corollary 3 straightforwardly to the following result, which generalizes [2, Theorem 31: (0 < ii Q sj, 1 2+s,+s,+.*.+sk. m

REFERENCES 1. E. R. BERLEKAMP, “Algebraic Coding Theory,” McGraw-Hill, New York, 1968. 2. C. R. P. HARTMANN AND K. K. TZENG, Generalizations of the BCH-bound, Iqform.and Control

20 (1972),

489-498.

F. J. MAC WILLIAMS AND N. J. A. SLOANE, “The Theory of Error-Correcting Codes,” North-Holland, Amsterdam, 1977. 4. W. W. PETERSON AND E. J. WELDON. JR.. “Error-Correcting Codes,” 2nd ed., MIT Press, Cambridge, Mass., (1972). 3.

Primed

in Belgium