A class of constacyclic BCH codes

Constacyclic codes are a subclass of linear codes and have been well studied. Constacyclic BCH codes are a family of constacyclic codes and contain BCH codes as a subclass. Compared with the in-depth study of BCH codes, there are relatively little study on constacyclic BCH codes. The objective of this paper is to determine the dimension and minimum distance of a class of q -ary constacyclic BCH codes of length q m − 1 q − 1 $\frac {q^{m}-1}{q-1}$ with designed distances δ i = q m − 1 − q ⌊ m − 3 2 ⌋ + i − 1 q − 1 $\delta _{i}=q^{m-1}-\frac {q^{\lfloor \frac {m-3}2 \rfloor +i }-1}{q-1}$ for 1 ≤ i ≤ min { ⌈ m + 1 2 ⌉ − ⌊ m q + 1 ⌋ , ⌈ m − 1 2 ⌉ } $1\leq i\leq \min \limits \{\lceil \frac {m+1}2 \rceil -\lfloor \frac {m}{q+1} \rfloor , \lceil \frac {m-1}2 \rceil \}$ . As will be seen, some of these codes are optimal.

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