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

Journal of Algebra Combinatorics Discrete Structures and Applications

article

Kazuki Kumegawa¹  Tatsuya Maruta² 

[1] Department of Mathematics and Information Sciences, Osaka Prefecture University;Department of Mathematical Sciences, Osaka Prefecture University

关键词: Optimal linear codes;    Griesmer bound;    Arcs in PG(r;    q);   

DOI  :  10.13069/jacodesmath.427968

学科分类：社会科学、人文和艺术（综合）

来源: Yildiz Technical University

【 摘 要 】

We prove the non–existence of [gq(4, d), 4, d]q codes for d = 2q 3 − rq2 − 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/qi 」 . 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 − rq2 − 2q + 1 ≤ d ≤ 2q 3 − rq2 − 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.

【 授权许可】

CC BY