A Characterization of Some Minihypers and its Application to Linear Codes

Abstract

Any {f,θr- 2+s; r,q}-minihyper includes a hyperplane in PG(r, q) if f≤θr-1 + s θ1 + q − 1 for 1 ≤ s ≤ q − 1, q ≥ 3, r ≥ 4, where θi = (qi + 1 − 1)/ (q − 1 ). A lower bound on f for which an {f, θr − 2 + 1; r, q}-minihyper with q≥ 3, r ≥ 4 exists is also given. As an application to coding theory, we show the nonexistence of [ n, k, n + 1 − qk − 2 ]q codes for k ≥ 5, q ≥ 3 for qk − 1 − 2q − 1 < n ≤ qk − 1 − q − 1 when k > q − $k > q - \sqrt q + 2$ and for $q^{k - 1} - q - \sqrt q - k + 2 < n \leqslant q^{k - 1} - q - 1$ when $k \leqslant q - \sqrt q + 2$, which is a generalization of [18, Them. 2.4].