On the geometric constructions of optimal linear codes

Abstract

In this paper we generalize the construction of Griesmer codes of Belov type. This leads to the construction of several codes of length g q (k, d) + 1, many of which are optimal. We also construct a q-divisible [q 2 + q, 5, q 2 − q] q code through projective geometry. As a projective dual of the code, we construct optimal codes, givingwhere n q (k, d) is the minimum length n for which an [n, k, d] q code exists.

On the minimum length of linear codes overF5

“…It is known that [182, 4, 145] 5 codes are unique up to equivalence, and M ′ C consists of Σ and an elliptic quadric, see [23]. Hence, M C − 2Σ gives a [26,4,20] 5 code, which has spectrum (a 1 , a 6 ) = (26, 130), and our assertion follows. Proof.…”

“…In this paper, we tackle the problem to determine n 5 (5, d) for all d. See [24] for the updated table of n 5 (5, d). The following results are already known for n 5 (k, d), see [1][2][3]5,7,9,11,21,20,25,24,30]. (3) n 5 (4, d) = g 5 (4, d) + 2 for d = 25.…”

On the Construction of LCD Codes overF5

“…From [17][18][19][20][21], we know that existence of many optimal linear codes over 5 F are determined. Yet, it is not known whether these codes are LCD codes.…”

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

“…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]. It is also known that n q (4, d) = g q (4, d) for d ≥ 2q 3 − 3q 2 + 1 for all q and that n q (4, d) = g q (4, d) + 1 for 2q 3 − 3q 2 − 2q + 1 ≤ d ≤ 2q 3 − 3q 2 for q ≥ 5 [12].…”