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

We present O(m3) algorithms for specifying the support of minimum-weight codewords of extended binary BCH codes of length n = 2m and designed distance d(m, s, i) := 2m-1-s-2m-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 22i-1-2i-1 elements, and a subspace of dimension m - 2i - s, specified by a basis. In some detail, for designed distance 6 ? 2j , j € {0, . . . ,m - 4}, we have a deterministic algorithm for even m ≥ 4, and a probabilistic algorithm with success probability 1 - O(2-m) for odd m > 4. For designed distance 28 ? 2j , j € {0, . . . ,m - 6}, we have a probabilistic algorithm with success probability ≥ 1 3 - O(2-m/2) for even m ≥ 6. Finally, for designed distance 120 ? 2j , j € {0, . . . , m-8}, we have a deterministic algorithm for m ≥ 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 € {0, . . . , ?m/2?}, and s ≤ 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 ≥ 2, 0 ≤ i ≤ ?m/2?, and 0 ≤ s ≤ 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 lowcomplexity 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 ? 2j , 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..