On the second generalized covering radius for binary primitive triple-error-correcting BCH codes

Official URL: https://dx.doi.org/10.1007/s10623-026-01903-0

Using methods from the theory of algebraic curves over finite fields, together with recent results on the arithmetic of cubic equations over finite fields, we obtain new upper bounds on the second generalized covering radius of binary primitive triple-error-correcting BCH codes. In particular we introduce the notion of weak second generalized covering radius R2(0)(BCH(3,m)), where BCH(3, m) is the binary primitive triple-error-correcting code of length 2m-1. This accounts almost all 2-dimensional F2-linear subspaces of F2n-3m. Among other results, we show that R2(0)(BCH(3,m))≤9 if m is odd and m≥11 (and if m is even and m≥20).