The non-existence of Griesmer codes with parameters close to codes of Belov type

Abstract

Hill and Kolev give a large class of q-ary linear codes meeting the Griesmer bound, which are called codes of Belov type (Hill and Kolev, Chapman Hall/CRC Research Notes in Mathematics 403, pp. 127-152, 1999). In this article, we prove that there are no linear codes meeting the Griesmer bound for values of d close to those for codes of Belov type. So we conclude that the lower bounds of d of codes of Belov type are sharp. We give a large class of length optimal codes with n q (k, d) = g q (k, d) + 1.

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

“…For k = 4, it is known that n q (4, d) = g q (4, d) for q 3 − q 2 − q + 1 ≤ d ≤ q 3 + q 2 + q, d ≥ 2q 3 − 3q 2 + 1 for all q and for 2q 3 − 5q 2 + 1 ≤ d ≤ 2q 3 − 5q 2 + 3q for q ≥ 7 ( [18,21]). The key contribution here is showing the non-existence of [g q (4, d), 4, d] q codes for many values of d close to these "Griesmer area", and it seems reasonable to seek a generalization for larger k. To this direction, see [3] and [4].…”

“…An [n, k, d] q code with generator matrix G is called extendable if there exists a vector h ∈ F k q such that the extended matrix [G, h T ] generates an [n + 1, k, d + 1] q code. The following theorems will be applied to prove that a [g q (3…”

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.…”