Efficient Algorithms for Constructing Minimum-Weight Codewords in Some Extended Binary BCH Codes

Abstract

We present $O(m^{3})$ algorithms for specifying the support of minimum-weight codewords of extended binary BCH codes of length $n=2^{m}$ and designed distance $d(m,s,i):=2^{m-1-s}-2^{m-1-i-s}$ for some values of $m,i,s$ , where m may grow to infinity. Here, the support is specified as the sum of two sets: a set of $2^{2i-1}-2^{i-1}$ elements, and a subspace of dimension $m-2i-s$ , specified by a basis. In some detail, for designed distance $6\cdot 2^{j}$ , $j\in \{0,\ldots ,m-4\}$ , we have a deterministic algorithm for even $m\geq 4$ , and a probabilistic algorithm with success probability $1-O(2^{-m})$ for odd $m\gt 4$ . For designed distance $28\cdot 2^{j}$ , $j\in \{0,\ldots , m-6\}$ , we have a probabilistic algorithm with success probability $\geq \frac {1}{3}-O(2^{-m/2})$ for even $m\geq 6$ . Finally, for designed distance $120\cdot 2^{j}$ , $j\in \{0,\ldots , m-8\}$ , we have a deterministic algorithm for $m\geq 8$ divisible by 4. We also show how Gold functions can be used to find the support of minimum-weight words for designed distance $d(m,s,i)$ (for $i\in \{0,\ldots ,\lfloor m/2\rfloor \}$ , and $s\leq m-2i$ ) whenever $2i|m$ . Our construction builds on results of Kasami and Lin, who proved that for extended binary BCH codes of designed distance $d(m,s,i)$ (for integers $m\geq 2$ , $0\leq i\leq \lfloor m/2\rfloor $ , and $0\leq s\leq m-2i$ ), the minimum distance equals the designed distance. The proof of Kasami and Lin makes use of a non-constructive existence result of Berlekamp, and a constructive “down-conversion theorem” that converts some words in BCH codes to lower-weight words in BCH codes of lower designed distance. Our main contribution is in replacing the non-constructive counting argument of Berlekamp by a low-complexity algorithm. In one aspect, the current paper extends the results of Grigorescu and Kaufman, who presented explicit minimum-weight codewords for extended binary BCH codes of designed distance exactly 6 (and hence also for designed distance $6\cdot 2^{j}$ , by a well-known “up-conversion theorem”), as we cover more cases of the minimum distance. In fact, we prove that the codeword constructed by Grigorescu and Kaufman is a special case of the current construction. However, the minimum-weight codewords we construct do not generate the code, and are not affine generators, except, possibly, for a designed distance of 6.