The aim of this work is a systematic investigation of the possible parameters of quasi-perfect (QP) binary and ternary linear codes of small dimensions and preparing a complete classification of all such codes. First, we give a list of infinite families of QP codes which includes all binary, ternary, and quaternary codes known to us. We continue further with a list of sporadic examples of binary and ternary QP codes. Later we present the results of our investigation where binary QP codes of dimensions up to 14 and ternary QP codes of dimensions up to 13 are classified.

For any integer q we present a new bound which is a q-ary combinatorial analog of the binary Grey-Rankin bound. For any prime power q we present two infinite classes of q-ary codes which meet this bound with integral equality. Moreover, we show how codes meeting this bound with equality are connected to several important classical combinatorial configurations, such as difference matrices and generalized Hadamard matrices.

We study a class of two-weight q-ary codes, considered by Baumert and McEliece, with regard to their properness or goodness in error detection. We prove for a large set of parameters that these codes are not good. For the remaining values of the parameters (which may or may not correspond to good codes) we determine if they are proper or not

We consider the q-ary analog of the binary Grey-Rankin bound, recently suggested by Fu, Kloeve and Shen. For any prime power q/spl ges/2, we give an infinite family of codes which reach this bound with equality. If the outer and inner codes are chosen as linear, a linear resulting code is obtained by the concatenation construction.

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.

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.

The undetected error probability of a linear code used to detect errors on a symmetric channel is a function of the symbol error probability /spl epsi/ of the channel and involves the weight distribution of the code. The code is proper, if the undetected error probability increases monotonously in /spl epsi/. Proper codes are generally considered to perform well in error detection. We show in this correspondence that maximum minimum distance (MMD) codes are proper.

Let n(k, d) be the smallest integer n for which a binary linear code of length n, dimension k, and minimum distance d exists. Using the residual code technique, the MacWilliams identities and the weight distribution of appropriate Reed-Muller codes, we prove that n(9, 64)=133, n(9, 120)/spl ges/244, n(9, 124)=252, and n(9, 184)=371. We also show that puncturing a known [322, 9, 160]-code yields length-optimal codes with parameters [319, 9, 158], [315, 9, 156], and [312, 9, 154].

Constructions of [162,8,80] and [159,8,78] codes are given. This solves the open problems of finding the minimum length of binary codes of dimension S and minimum distances 78 and 80, respectively.

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.

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.

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.>

It is shown that the covering radius of any binary linear (14, 6) code containing the all-one vector is at least 4. Since the minimum covering radius of a binary linear (14, 6) code is 3, this shows that in general the minimum of the covering radius is not achieved by codes containing the all-one vector.>

Letn(k,d)be the smallest integernsuch that a binary linear code of lengthn, dimensionk, and minimum distance at leastdexists. New results are given that improve the best previously known bounds onn(8,d).