Shortened finite geometry codes (Corresp.)

A method of shortening finite analytic geometry codes, projective-geometry (PG) codes, Euclidean-geometry (EG) codes, and 2-fold EG codes is presented. The shortened codes preserve the feature of being majority-logic decodable and they have the same error-correcting capability as the original codes. Combinatorial expressions for the parity-check symbols of the shortened codes are derived.     

A characterization of N-weight projective codes (Corresp.)

Using the Fourier transform for vector spaces over the finite field GF(p^{m}), a characterization ofN-weight linear codes is given. Some particular cases containing Camion's relations for two parameters' difference sets are examined, and a construction using the notion of normal spread is given.     

On the minimum distance of certain reversible cyclic codes (Corresp.)

The minimum distance of a class of reversible cyclic codes has been proved to be greater than that given by the BCH bound. It is also noted that this class of codes includes the class of primitive double-error-correcting binary codes of Melas as well as the class of nonprimitive double-error-correcting binary codes discovered by Zetterberg as special cases.     

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 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.     

Permutation decoding of certain triple-error-correcting binary codes (Corresp.)

In a recent note [1] an analysis of certain binary double-error-correcting cyclic codes was made from the point of view of permutation decodability. Specifically, upper bounds on the rates of these codes were established so that the codes would be permutation decodable. The object of the present correspondence is to give corresponding results for binary triple-error-correcting cyclic codes of certain lengths.     

Note on majority-logic decoding of finite geometry codes (Corresp.)

In a recent paper [1], techniques for reducing the number of majority-logic decoding steps for finite geometry codes have been proposed. However, the lower bound of [1, lemma 4] is incorrect; finite geometry codes, in general, cannot be decoded in less than or equal to three steps of orthogonalization, as was claimed. This correspondence presents a decoding procedure for finite geometry codes that requires as few decoding steps as possible. It is shown that the minimum number of steps is a logarithmic function of the dimension of the associated geometry.     

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.     

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 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 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.     

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.