Problems of Information Transmission, Vol. 40, No. 2, 2004, pp. 135–146. Translated from Problemy Peredachi Informatsii, No. 2, 2004, pp. 37–49.
Original Russian Text Copyright
2004 by Daskalov, Metodieva.
The Nonexistence of Ternary [284, 6, 188] Codes
R. Daskalov and E. Metodieva
Department of Mathematics, Technical University of Gabrovo, Bulgaria
Received August 20, 2003; in ﬁnal form, January 8, 2004
Abstract—Let [n, k, d]
codes be linear codes of length n,dimensionk, and minimum Ham-
ming distance d over GF (q). Let n
(k, d) be the smallest value of n for which there exists an
[n, k, d]
code. It is known from [1, 2] that 284 ≤ n
(6, 188) ≤ 285 and 285 ≤ n
(6, 189) ≤ 286.
In this paper, the nonexistence of [284, 6, 188]
codes is proved, whence we get n
(6, 188) = 285
(6, 189) = 286.
Let GF (q) denote the Galois ﬁeld of q elements, and let V (n, q) denote the vector space of all
ordered n-tuples over GF (q). The Hamming weight of a vector x, denoted by wt(x), is the number
of nonzero entries in x. A linear code C of length n and dimension k over GF (q)isak-dimensional
subspace of V (n, q). Such a code is called an [n, k, d]
code if its minimum Hamming distance is d.
For a linear code, the minimum distance is equal to the smallest of weights of nonzero codewords.
A central problem in coding theory is that of optimizing one of the parameters n, k,andd for
given values of the other two and q ﬁxed. Two versions are:
Problem 1. Find d
(n, k), the largest value of d for which there exists an [n, k, d]
Problem 2. Find n
(k, d), the smallest value of n for which there exists an [n, k, d]
A code which attains one of these two values is called optimal.
Systematic research of ternary optimal codes was initiated in , where the values of n
k ≤ 4 for all d and the values of n
(5,d) for all but 30 values of d were found. Five optimal codes
were constructed in [4,5], one in , and three optimal codes were constructed in . Essentials of
the remaining nonexistence cases were settled in a series of papers [5, 8, 9] and also in .
The ﬁrst results and tables for n
(6,d) were published in [11, 12]. After that, the state of
knowledge for n
(6,d), d ≤ 243, was summarized in . Many new ternary codes of dimension
k = 6 were obtained in [14,15]. New nonexistence results were obtained in [16–23]. An updated table
(6,d) was presented in . Recently, the nonexistence of [69, 6, 44]
,[81, 6, 52]
, [108, 6, 70]
[157, 6, 103]
, [257, 6, 170]
, and [269, 6, 178]
codes was proved , and a new table for n
was presented (see ).
2. PRELIMINARY RESULTS
A well-known lower bound for n
(k, d) is the Griesmer bound [25, 26]
(k, d) ≥ g
(x denotes the smallest integer ≥ x).
Supported in part by the National Science Fund of Bulgaria under contract with TU–Gabrovo.
2004 MAIK “Nauka/Interperiodica”