The \(\ell \) -th Schur power \(\mathcal {C}^{\ell }\) of a linear code \(\mathcal {C}\) plays an important role in solving some cryptographic problems. For a positive integer m, let \(\mathcal {C}(\delta )\) be the primitive narrow-sense Bose-Chaudhuri-Hocquenghem (BCH) code of length \(n=q^m-1\) over \(\mathbb {F}_q\) . We shall focus on the Schur powers of primitive narrow-sense BCH codes \(\mathcal {C}(\delta )\) due to their elegant algebraic structures and wide applications. In this paper, the parameters of the powers of a class of BCH codes will be explored. It is known that the Schur square \(\mathcal {C}^{2}\) is the product of a linear code and itself, while the Schur cube \(\mathcal {C}^{3}\) is the product of two distinct codes \(\mathcal {C}^{2}\) and \(\mathcal {C}\) . Then the Schur power \(\mathcal {C}^{\ell }\) can be recursively defined by \(\mathcal {C}^{\ell }=\mathcal {C}^{\ell -1} \star \mathcal {C}\) . In this sense, it is very important to study the two fundamental cases: the Schur square and cube. The Schur squares of cyclic codes and primitive BCH codes were explored in [6, 21], respectively. Motivated by these results, we investigate the Schur cubes of primitive narrow-sense BCH codes \(\mathcal {C}(\delta )\) in this paper. We will present a necessary and sufficient condition to guarantee that \(\mathcal {C}^3(\delta ) \ne \mathbb {F}_q^n\) by giving restrictions on the designed distance \(\delta \) , where \(2 \le \delta \le n\) . The dimensions and lower bounds on the minimum distances of \(\mathcal {C}^3(\delta )\) are investigated in some cases. Several optimal codes can be found. Moreover, we present a class of [n, k, d] cyclic codes over \(\mathbb {F}_q\) with \(k\ge \frac{n}{2}\) and \(d\ge \sqrt{n}\) (or \(d\ge \sqrt{n}-1\) ) via the Schur power.
