A Necessary and Sufficient Condition for the Existence of Some Ternary [n, k, d] Codes Meeting the Greismer Bound

Abstract

Let k and d be any integers such that k ≥ 4 and $$ . Then there exist two integers α and β in {0,1,2} such that $$. The purpose of this paper is to prove that (1) in the case k ≥5 and (α,β) = (0,1), there exists a ternary $\left[ {n,k,d} \right]$ code meeting the Griesmer bound if and only if $3^{k - 1} - 3^{k - 2} - 2 \leqslant d \leqslant 3^{k - 1} - 3^{k - 2}$ and (2) in the case k ≥4 and (α,β) = (0,2) or (1,1), there is no ternary $\left[ {n,k,d} \right]$ code meeting the Griesmer bound for any integers k and d and (3) in the case k ≥5 and $d = 3^{k - 1} - 3^{\lambda + 1} - 3^{k - 2}$, there is no projective ternary $[g_3 (k,d) + 1,k,d]$ code for any integers k and λ such that 1≤λ≤k-3, where ${\text{g}}_{\text{3}} (k,d) = \upsilon _k - \upsilon _{\lambda + 2} - \upsilon _{k - 1}$ and $\upsilon _i = (3^i - 1)/(3 - 1)$ for any integer i ≥0. In the special case k=6, it follows from (1) that there is no ternary linear code with parameters [233,6,154] , [234,6,155] or [237,6,157] which are new results.