On the Minimum Length of some Linear Codes of Dimension 5

One of the interesting problems in coding theory is to determine the valuenq(k, d) which denotes the smallest number n such that an [n, k,d]q code existsfor given k, d and q.For k ≤ 5, there are many results for q ≤ 5 and for k = 6, the results areconcentrated in the ternary code ([1],[3]). In this talk, we concentrated in theproblem to find the exact value nq(6, d). It is known that nq(6, d) = gq(6, d)for q−q−q+1 ≤ d ≤ q and q−q−q+1 ≤ d ≤ q−q for any q. Marutaproved that nq(6, d) = gq(6, d) + 1 for q 5 − q − 2q + 1 ≤ d ≤ q − q − q ([2]).For q − q − q − q + 1 ≤ d ≤ q − q − q, we prove non-existence of a[gq(6, d), 6, d]q code and we construct a [gq(6, d)+1, 6, d]q code by constructingappropriate 0-cycle in the projective space. Thus we get the minimum lengthnq(6, d) = gq(6, d) + 1 for q 5 − q − q − q + 1 ≤ d ≤ q − q − q. References[1] N. Hamada and T. Helleseth, The nonexistence of some ternary linear codesand update of the bound for n3(6, d), 1 ≤ d ≤ 243, Mathematica Japonica, 52,No.1 (2000), 31–43.[2] T. Maruta, On the nonexistence of q-ary linear codes of dimension five,Designs, Codes and Cryptography 22 (2001), 165–177.[3] T. Maruta, Griesmer Bound for Linear Codes over Finite Fields. [Online].Available: http://www.appmath.osaka-wu.ac.jp/ maruta/griesmer.htm