Let $K_3(7,1)$ denote the minimum cardinality of a ternary code of length $7$ and covering radius one. In a previous paper, we improved on the lower bound $K_3(7,1)\ge 147$ by showing that $K_3(7,1)\ge 150$. In this note, we prove that $K_3(7,1)\ge 153$.

We denoted by nq(k, d), the smallest value of n for which an [n, k, d]q code exists for given q, k, d. Since nq(k, d) = gq(k, d) for all d ≥ dk + 1 for q ≥ k ≥ 3, it is a natural question whether the Griesmer bound is attained or not for d = dk , where gq(k, d) = ∑[d/q^i], i=0,...,k-1, dk = (k − 2)q^(k−1) − (k − 1)q^(k−2). It was shown by Dodunekov [2] and Maruta [9], [10] that there is no [gq(k, dk ), k, dk ]q code for q ≥ k, k = 3, 4, 5 and for q ≥ 2k − 3, k ≥ 6. The purpose of this paper...