Parameters of Squares of Primitive Narrow-Sense BCH Codes and Their Complements

Studying the Schur square of a linear code is an important research topic in coding theory. Schur squares have important applications in cryptography and private information retrieval schemes, notably in secure multiparty computing or designing bilinear multiplication algorithms in finite extensions of finite fields through the notion of supercodes. Thanks to their exciting applications in cryptography, squares and powers of several linear codes have been investigated. In this paper, we will focus on the Schur square of a relevant well-known subclass of cyclic codes, Bose-Chaudhuri-Hocquenghem codes (BCH codes), which have wide applications in communication and storage systems and benefit from explicit defining sets that include consecutive integers, which gives the advantage of analyzing the parameters of BCH codes and their complements. Our main objective is to investigate the parameters of the squares of primitive narrow-sense BCH codes
$\mathcal C(\delta)$
and their complements
$\mathcal C(\delta)^{c}$
. We will present two sufficient and necessary conditions to guarantee that
$\mathcal C^{2}(\delta) \ne \Bbb F_{q}^{n}$
and
$\mathcal C^{2}(\delta)^{c} \ne \Bbb F_{q}^{n}$
by giving restrictions on designed distance
$\delta $
, where
$2 \le \delta \le n$
. Based on these two characterizations, the dimensions and minimum distances of
$\mathcal C^{2}(\delta)$
and
$\mathcal C^{2}(\delta)^{c}$
are investigated in some cases. The dimensions of these squares are determined explicitly, and lower bounds on the minimum distance are given.