On cyclic MDS codes of length q over GF(q) (Corresp.)

It is shown that a cyclic codeCof lengthqover GF(q)is the maximum distance separable if and only if either1) qis a prime, in which caseCis equivalent, up to a coordinate permutation, to an extended Reed-Solomon code, or2) Cis a trivial code of dimensionk in {1, q - 1, q }. Hence there exists a nontrivial cyclic extended Reed-Solomon code of lengthqover GF(q)if and only ifqis a prime.     

On MDS codes via Cauchy matrices

A special form of Cauchy matrix is used to obtain a tighter bound for the validity region of the maximum distance separable (MDS) conjecture and a new compact characterization of generalized Reed-Solomon codes. The latter is further used to obtain constructions and some existence results for long [2k, k] double-circulant MDS codes     

On simplex-type binary codes (Corresp.)

The rank of the generating matrix in certain cases is computed. The result is perhaps of interest in its own right, apart from any applications it may have. The method of proof, utilizing as it does a knowledge of the parameters of a certain association scheme, is a technique which can be profitably used elsewhere.     

On maximal comma-free codes (Corresp.)

In this correspondence a binary maximal comma-free code for words of length 10 and the search technique used to construct it are described.     

On maximum-distance-separable convolutional codes (Corresp.)

We define maximum-distance-separable convolutional codes as systematic codes with (feedback) minimum distance exceeding the number of check digits in a constraint length. The maximum length of such codes is determined for certain small fields when the rate isfrac{1}{2}.     

On codes overZ_m(Corresp.)

The factorization of Abelian codes overZ_{m}, i.e., ideals inZ_{m},G,Ga finite Abelian group, corresponding to a factorization ofmand that of G as a product of cyclic groups are considered. Quasi-Abelian codes overZ_{m}are considered and it is shown that every quasi-Abelian code overZ_{m}is a direct sum of Abelian codes overZ_{m}. A factorization of Gray codes overZ_{m}is also considered.     

On certain projective geometry codes (Corresp.)

LetVbe an(n, k, d)binary projective geometry code withn = (q^{m}-1)/(q - 1), q = 2^{s}, andd geq [(q^{m-r}-1)/(q - 1)] + 1. This code isr-step majority-logic decodable. With reference to the GF(q^{m}) = {0, 1, alpha , alpha^{2} , cdots , alpha^{n(q-1)-1} }, the generator polynomialg(X), ofV, hasalpha^{nu}as a root if and only ifnuhas the formnu = i(q - 1)andmax_{0 leq l < s} W_{q}(2^{l} nu) leq (m - r - 1)(q - 1), whereW_{q}(x)indicates the weight of the radix-qrepresentation of the numberx. LetSbe the set of nonzero numbersnu, such thatalpha^{nu}is a root ofg(X). LetC_{1}, C_{2}, cdots, C_{nu}be the cyclotomic cosets such thatSis the union of these cosets. It is clear that the process of findingg(X)becomes simpler if we can find a representative from eachC_{i}, since we can then refer to a table, of irreducible factors, as given by, say, Peterson and Weldon. In this correspondence it was determined that the coset representatives for the cases ofm-r = 2, withs = 2, 3, andm-r=3, withs=2.     

On synchronizable binary cyclic codes (Corresp.)

In this correspondence a method is presented whereby the average synchronization-error-correcting capability of Tavares' subset codes may be improved with no additional cost in rate and with only a small increase in the complexity of encoding and decoding. The method consists simply in shifting every word of the subset codes in such a way so that the shifted versions have a maximum number of leading and trailing zeros. A lower bound on the increase in synchronization-error-correcting capability provided by this method is derived.     

On t-constant codes and designs (Corresp.)

LetCbe a binary code containing the zero word, with fundamental parametersd,s,d', ands, and lettbe a nonnegative integer such that the words of any fixed weight ofCform at-design, withtmaximal in this respect. It is shown thatt geq max(d-s',d'-s). Moreover, ifCcontains the all one word, thent geq d'- s+1and ifCis an even weight code thent geq d - s' + 1. Finally, the following inequality is derived:d'-s < d, provided thatd>2.     

On binary majority-logic decodable codes (Corresp.)

LetVprimebe a binary(n,k)majority-logic decodable code withgprime (X)as its generator polynomial and odd minimum distanced. LetVbe the(n, k - 1)subset code generated bygprime (X)(1 + X). This correspondence shows thatVis majority-logic deeodable withd + 1orthogonal estimates. This fact is useful in the simultaneous correction of random errors and erasures.     

Optimal codes (Corresp.)

Some known results on the nonexistence of linear optimal codes can be very easily proved using results on the weight distribution of such codes.     

Duadic codes (Corresp.)

The notion of duadic codes over GF(2)is generalized to arbitrary fields. Duadic codes of composite length are constructed. An upper bound is given for the minimum distance of duadic codes of length a prime power.     

Multigram codes (Corresp.)

A multigram code is a list of codewords for multigrams (of various lengths) belonging to a setS. Three interrelated problems require designing a good setS, good codewords, and a good strategy for dissecting messages into multigrams ofS. Focus is placed on the dissection strategy and applications to English.