Stefan M. Dodunekov

An improvement of the Griesmer bound for some small minimum distances

Abstract In this paper we give some lower and upper bounds for the smallest length n ( k , d ) of a binary linear code with dimension k and minimum distance d . The lower bounds improve the known ones for small d . In the last section we summarize what we know about n (8, d ).

On the cyclic redundancy-check codes with 8-bit redundancy

Polynomials of degree eight over GF(2) which are suitable to be used as generator polynomials for cyclic redundancy-check (CRC) codes are investigated. Their minimum distance, properness and undetected error probability for binary symmetric channels (BSCs) are compared with the existing ATM standard. Covering radii and the weight distributions of the leaders of cosets are given for all examined codes.

Binary self-dual codes with automorphisms of composite order

In this paper, we present some results concerning a decomposition of binary self-dual codes having an automorphism of an order which is the product of two odd prime numbers. These results are applied to construct self-dual [72,36,12] codes with an automorphism of order 15. Furthermore, it is proved that the automorphism group of a putative binary extremal self-dual [72,36,16] code contains at most 28 nontrivial types of automorphisms of odd order. A complete list of these possible types of automorphisms is presented.

New bounds on binary linear codes of dimension eight (Corresp.)

Let n(k,d) be the smallest integer n such that a binary linear code of length n , dimension k , and minimum distance at least d exists. New results are given that improve the best previously known bounds on n(8,d) .

On the [28, 7, 12] binary self-complementary codes and their residuals

Recently Jungnickel and Tonchev have shown that there exist at least four inequivalent binary selfcomplementary [28, 7, 12] codes and have asked if there are other [28, 7] codes with weight distributionA0=A28=1,A12=A16=63. In the present paper we give a negative answer: these four codes are, up to equivalence, the only codes with the given parameters. Their residuals are also classified.

Sufficient conditions for good and proper error-detecting codes

The performance of linear block codes over a finite field is investigated when they are used for pure error detection. Sufficient conditions for a code to be good or proper for error detection are derived.

On the performance of the ternary [13, 7, 5] quadratic-residue code

We investigate the weight structure and error-correcting performance of the ternary [13, 7, 5] quadratic-residue code. It is shown that the covering radius of the code is equal to three and it is t-proper for error correction for any t=0, 1, 2. Two decoding algorithms are suggested.

A survey on proper codes

The performance of a linear t-error correcting code over a q-ary symmetric memoryless channel with symbol error probability @e is characterized by the probability that a transmission error will remain undetected. This probability is a function of @e involving the code weight distribution and the weight distribution of the cosets of minimum weight at most t. When the undetectable error probability is an increasing function of @e, the code is called t-proper. The paper presents sufficient conditions for t-properness and a list of codes known to be proper, many of which have been studied by these sufficient conditions. Special attention is paid to error detecting codes of interest in modern communication.

Algebraic decoding of the Zetterberg codes

The Zetterberg codes are one of the best known families of double-error correcting binary linear codes. Unfortunately, no satisfactory decoding algorithm has been known for them until recently when an algebraic decoding algorithm was described by P. Kallquist (1989). It requires, however, to solve a quadratic equation in order to decide whether 2 or 3 errors have occurred. A simple criterion is derived to determine whether 1, 2, or 3 errors have occurred when a Zetterberg code is used for data transmission. Based on criterion a new decoding algorithm is proposed which is faster than the known one. >

Almost-MDS and near-MDS codes for error detection

The error detection capability of almost-MDS (AMDS) and nearly-MDS (NMDS) codes is studied. Necessary and sufficient conditions for the codes to be proper or good for error detection are given.