BCH Codes with Minimum Distance Proportional to Code Length

BCH codes are among the best practical cyclic codes widely used in consumer electronics, communication systems, and storage devices. However, not much is known about BCH codes with large minimum distance. In this paper, we consider narrow-sense BCH codes of length $n = \frac{q^m-1}{N}$ with designed distance $\delta = \frac{s}{q-1}n$ proportional to $n$, where $N$ divides $\frac{q^m-1}{q-1}$ and $1 \le s \le q-1$. We determine both their dimensions and minimum distances. In particular, when $N=1$, the codes are primitive, with minimum distance $d=\frac{s}{q-1}(q^m-1)$ and dimension $k = (q-s)^m$. The general result on code dimensions is achieved by applying generating functions and inverse discrete Fourier transforms to an enumeration problem.