On the minimum length of some linear codes

Abstract

We determine the minimum length n q (k, d) for some linear codes with k ≥ 5 andAMS Classifications 94B65 · 94B05 · 51E20 · 05B25

A class of optimal linear codes of length one above the Griesmer bound

“…Proof (1) For s = 2, we can see its proof in [2]. When s ≥ 3, by combining Theorems A and B, we have n q (k, d) ≥ g q (k, d) + 1 for q ≥ 2s − 1 and k ≥ 2s + 1.…”

“…When s = 1 we have n q (k, d) = g q (k, d) + 1 for q ≥ 3 and k > q − √ q + 2 (see [13]), and when s = 2 we also have n q (k, d) = g q (k, d) + 1 for q ≥ 3 and k ≥ 5 (see [2]). For the case s = k − 2, since Klein [10] proved the nonexistence of Griesmer codes, we note n q (k, d) ≥ g q (k, d) + 1 for q ≥ 2k − 3 and k ≥ 4.…”

Non-existence of some 4-dimensional Griesmer codes over finite fields

“…Theorem 1.1 ( [12]). There exists no [g q (4 As a continuation on the non-existence of Griesmer codes for k = 4, we prove the following four theorems. [10].…”

“…Since the existence of an [n, k, d] q code implies the existence of an [n − 1, k, d − 1] q code by puncturing, we get the following results from Theorems 1.2-1.5. Corollary 1.6. n q (4…”

On the geometric constructions of optimal linear codes

“…Most of the known optimal [n, k, d] q codes have length n = g q (k, d) or n = g q (k, d) + 1, see [5], [6], [7], [18], [19], but the systematic method to construct codes of length g q (k, d) + 1 is unknown. The purpose of this paper is to give a method to construct codes over F q whose lengths are close to the Griesmer bound.…”