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

Abstract

We prove the non-existence of [gq(4, d), 4, d]q codes for d = 2q 3 − rq 2 − 2q + 1 for 3 ≤ r ≤ (q + 1)/2, q ≥ 5; d = 2q 3 − 3q 2 − 3q + 1 for q ≥ 9; d = 2q 3 − 4q 2 − 3q + 1 for q ≥ 9; and d = q 3 − q 2 − rq − 2 with r = 4, 5 or 6 for q ≥ 9, where gq(4, d) = 3 i=0 d/q i. This yields that nq(4, d) = gq(4, d) + 1 for 2q 3 −3q 2 −3q+1 ≤ d ≤ 2q 3 −3q 2 , 2q 3 −5q 2 −2q+1 ≤ d ≤ 2q 3 −5q 2 and q 3 −q 2 −rq−2 ≤ d ≤ q 3 −q 2 −rq with 4 ≤ r ≤ 6 for q ≥ 9 and that nq(4, d) ≥ gq(4, d) + 1 for 2q 3 − rq 2 − 2q + 1 ≤ d ≤ 2q 3 − rq 2 − q for 3 ≤ r ≤ (q + 1)/2, q ≥ 5 and 2q 3 − 4q 2 − 3q + 1 ≤ d ≤ 2q 3 − 4q 2 − 2q for q ≥ 9, where nq(4, d) denotes the minimum length n for which an [n, 4, d]q code exists.

On the Minimum Length of Linear Codes over the Field of 9 Elements

“…It follows from (2.3)-(2.5) that C has spectrum (a 8 , a 17 , a 26 ) = (36, 32, 752). Let δ be a 17-plane, whose spectrum (τ 0 , τ 1 , τ 2 , τ 3 ) is one of the four possible spectra for [17,3,14] 9 codes in Table 1. Let c 17 be the number of 17-planes ( = δ) through a fixed t-line on δ.…”

“…We have γ 0 = 1 by Lemma 2.1. Let ∆ be a γ 2 -plane, whose spectrum is one of the four spectra for [17,3,14] 9 codes in Table 1. We have a i = 0 for all i / ∈ {0, 1, 7, 8, 9, 10, 16, 17} by the first sieve.…”

“…For k = 3, n q (3, d) is known for all d for q 9. In this paper, we tackle the problem to determine n 9 (4, d) for all d. See [25] for the updated table of n q (k, d) for some small q and k. The following results are already known for n 9 (k, d) with k = 3, 4, see [5,14,15,16,17,19,21,24,25,27]. Theorem 1.5 ( [21,24]).…”

“…9 ,[17,3,14] 9 ,[16,4,12] 9 ,[24,4,19] 9 , [36, 4, 30] 9 , [41, 4, 34] 9 , [48, 4, 40] 9 , [58, 4, 49] 9 , [92, 4, 80] 9 , [102, 4, 88] 9 , [109, 4, 94] 9 , [118, 4, 102] 9 , [125, 4, 108] 9 and [130, 4, 112] 9 . Let C be the [132, 4, 114] 9 code with generator matrix G = [5510 13 ] + 1017 13 + 1215 13 + 1031 13 + 1167 13 + 1211 13 + 1705 13 + 1382 13 + 1706 13 + 1334 13 + 1051 1 + 1051 1 .…”

On optimal linear codes of dimension 4

“…(b) The nonexistence of a [g q (4, d), 4, d] q code for d = 2q 3 − rq 2 − q + 1 for 3 ≤ r ≤ q − q/p, q = p h with p prime, is proved in [19]. We conjecture that a [g q (4, d), 4, d] q code for d = 2q 3 − rq 2 − q + 1 with r = q − q/p − 1 does not exist for non-prime q ≥ 8, which is valid for q = 8, 9 by Theorem 1.5 and [17].…”