Products of Some Primitive BCH Codes and Their Complements

Schur product was originally proposed in coding theory for algebraic decoding algorithms and widely applied to solve some cryptographic problems in recent years. This shows the great importance of the Schur product in both coding theory and cryptography. As a well-known subclass of cyclic codes, Bose-Chaudhuri-Hocquenghem codes (BCH codes) have wide applications in communication and storage systems. Let
$\mathcal {C}_{1}$
and
$\mathcal {C}_{2}$
be two primitive BCH codes over
$\Bbb {F}_{q}$
with designed distances
$\delta _{a}$
and
$\delta _{b}$
, respectively, where
$2 \leq \delta _{a}, \, \delta _{b} \leq n$
. Let
$\mathcal {C}_{1}^{c}$
and
$\mathcal {C}_{2}^{c}$
be the complements of
$\mathcal {C}_{1}$
and
$\mathcal {C}_{2}$
, respectively. This paper aims to investigate the parameters of the products
$\mathcal {C}_{1} \star \mathcal {C}_{2}$
and
$\mathcal {C}_{1}^{c} \star \mathcal {C}_{2}^{c}$
. We will present some sufficient and necessary conditions to guarantee that
$\mathcal {C}_{1} \star \mathcal {C}_{2} \neq \Bbb F_{q}^{n}$
and
$\mathcal {C}_{1}^{c} \star \mathcal {C}_{2}^{c} \neq \Bbb F_{q}^{n}$
by giving restrictions on the designed distances
$\delta _{a}$
and
$\delta _{b}$
of the two BCH codes, respectively. The dimensions of these products are determined explicitly and lower bounds on the minimum distance are developed in some cases. Some optimal or best known codes are found. Moreover, it should be emphasized that a class of
$[n, k, d]$
cyclic codes over
$\Bbb F_{q}$
with dimension
$k \ge \frac {n}{2}$
and
$d \ge \sqrt {n}$
are presented.