Construction of I-Deletion-Correcting Ternary Codes

Finding large deletion correcting codes is an important issue in coding theory. Many researchers have studied this topic over the years. Varshamov and Tenegolts constructed the Varshamov-Tenengolts codes (VT codes) and Levenshtein showed the Varshamov-Tenengolts codes are perfect binary one-deletion correcting codes in 1992. Tenegolts constructed T codes to handle the non-binary cases. However the T codes are neither optimal nor perfect, which means some progress can be established. Latterly, Bours showed that perfect deletion-correcting codes have a close relationship with design theory. By this approach, Wang and Yin constructed perfect 5-deletion correcting codes of length 7 for large alphabet size. For our research, we focus on how to extend or combinatorially construct large codes with longer length, few deletions and small but non-binary alphabet especially ternary. After a brief study, we discovered some properties of T codes and produced some large codes by 3 different ways of extending some existing good codes.

Ternary Barker codes

Certain ternary codes having good autocorrelation properties akin to Barker codes are described.

New Bounds for the Maximum Size of Ternary Constant Weight Codes

This work was partially supported by the Bulgarian National Science Fund under Grant I–618/96.

Special subsets and optimum codes based on lattices and cosets

This paper introduces the concept of special subsets when applied to generator matrices based on lattices and cosets as presented by Calder-bank and Sloane. By using the special subsets we propose a non exhaustive code search for optimum codes. Although non exhaustive, the search always results in optimum codes for given (k1, V, Λ/Λ′). Tables with binary and ternary optimum codes to partitions of lattices with 8, 9 e 16 cosets, were obtained.

On the Error-Correcting Performance of some Binary and Ternary Linear Codes

In this work, we determine the coset weight spectra of all binary cyclic codes of lengths up to 33, ternary cyclic and negacyclic codes of lengths up to 20 and of some binary linear codes of lengths up to 33 which are distance-optimal, by using some of the algebraic properties of the codes and a computer assisted search. Having these weight spectra the monotony of the function of the undetected error probability after t-error correction P(t)ue (C,p) could be checked with any precision for a linear time. We have used a programm written in Maple to check the monotony of P(t)ue (C,p) for the investigated codes for a finite set of points of p € [0, p/(q-1)] and in this way to determine which of them are not proper.

Necessary and Sufficient Conditions for the Extendability of Ternary Linear Codes

We give the necessary and sufficient conditions for the extendability of ternary linear codes of dimension k ≥ 5 with minimum distance d ≡ 1 or 2 (mod 3) from a geometrical point of view.

A formalized optimum code search for q-ary trellis codes

Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES)

The Nonexistence of [132, 6, 86]3 Codes and [135, 6, 88]3 Codes

We prove the nonexistence of [g3(6, d), 6, d]3 codes for d = 86, 87, 88, where g3(k, d) = ∑⌈d/3i⌉ and i=0 ... k−1. This determines n3(6, d) for d = 86, 87, 88, where nq(k, d) is the minimum length n for which an [n, k, d]q code exists.