Two Families of BCH Codes and New Quantum Codes

Hermitian dual-containing constacyclic BCH codes and related quantum codes of length $\frac{q^{2m}-1}{q+1}$

“…We found that some optimal or almost optimal codes can be obtained from the narrowsense constacyclic BCH codes of length n = q 2m −1 q+1 with special values of q and m (see Remark 2 in Subsection 3.2). In addition, some quantum codes constructed from narrowsense constacyclic BCH codes of length n = q 2m −1 q+1 have better parameters compared to some known results studied in the literature [2,17,18,24,25,35]. Thus, this family of constacyclic BCH codes deserve to be investigated further.…”

“…Then they obtained some new quantum codes with minimum distance d > 2q 2 . Wang et al [35] focused on a class of q 2 -ary constacyclic BCH codes of length n = 2(q m + 1) with odd q and any m ≥ 3, via Hermitian construction, they gained many new quantum codes, which are not covered in the literature.…”

“…The main reason is that the union of q 2 -cyclotomic cosets for the defining set of constacyclic BCH codes will involve more repeated ingredients as the size of finite field and the design distance become larger. Very recently, Wang et al [35] study Hermitian dualcontaining narrow-sense constacyclic BCH codes of length q 2m −1 ρ and aim at constructing new quantum codes with the corresponding constacyclic BCH codes, where ρ|(q + 1), and m ≥ 3. By employing the technique of [38], they determine all coset leaders of cyclotomic cosets in the defining set and discuss the q-adic expansion about the design distance to compute the dimension of constacyclic BCH codes.…”

“…By employing the technique of [38], they determine all coset leaders of cyclotomic cosets in the defining set and discuss the q-adic expansion about the design distance to compute the dimension of constacyclic BCH codes. This method is efficient for the case of odd m (see Theorem 3 in [35]), however, the derived parameters of constacyclic BCH codes and the related quantum codes are not concise enough for the case of even m ≥ 4 (see Theorem 4 in [35]).…”

“…Thus, this family of constacyclic BCH codes deserve to be investigated further. In this article, we focus on determining the parameters of Hermitian dual-containing constacyclic BCH codes and the related quantum codes of length n = q 2m −1 q+1 for the case of even m, and the results about the case of odd m can be found in Subsection 2.2 of [24] or Theorem 3 in [35]. Note that in the case of even m, it is more challenging to derive the parameters of this family of Hermitian dual-containing constacyclic BCH codes, since the defining set of the considered codes involves a more intricate structure compared with the case of odd m. We provide necessary and sufficient conditions for the existence of narrow-sense Hermitian dual-containing constacyclic BCH codes (see Theorem 3), which allows one to identify all constacyclic BCH codes that can be used in quantum codes construction.…”