On the minimum length of linear codes overF5

Abstract

a b s t r a c tWe construct a lot of new [n, 5, d] 5 codes close to the Griesmer bound and prove the nonexistence of some Griesmer codes to determine the exact value of n 5 (5, d) or to improve the known upper bound on n 5 (5, d), where n q (k, d) is the minimum length n for which an [n, k, d] q code exists. We also give the updated table for n 5 (5, d) for all d except some known cases.

On optimal linear codes of dimension 4

“…It is shown in [28] that many good codes can be constructed from orbits of projectivities. (3,2). Then, the matrix [g 7 ] generates a cyclic Hamming [7, 4, 3] 2 code and the matrix…”

“…) is (c 23 , c 26 ) =(3,5). Hence we get 4259 = (LHS of (11)) ≤ 9 × 73 + 325 = 982, which contradicts(11).…”

“…We have a 1 = 0 from Lemma 2.1(e) since ∆ 1 has no 1-line. If a 14-plane δ exists, it follows from (10) that M(δ) gives a [14,3,12] 8 code, which does not exist. In this way, using Theorem 4.2 and Lemma 2.1, one can get a i = 0 for all i ∈ {0, 7-10, 15, 23-26}.…”

“…Hence, C is extendable by Theorem 2.8. Assume that adding a point P to the multiset M C gives a 101-plane δ corresponding to a [101, 3,88] Proof. Let C be a putative Griesmer [790, 4, 690] 8 code.…”

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

“…When ∆ has spectrum (E), we have 81λ 2 42615, i.e., λ Let C be a [78, 3, 69] 9 code. Since C is Griesmer, the set C 0 of 0-points for C forms a (13, 1)-blocking set in PG (2,9). If C 0 contains a line l, then C 0 consists of l and three points, say Q 1 , Q 2 , Q 3 .…”