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

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-2(i-s) , specified by a basis. In some detail, for designed distance 6 & sdot;2j , j is an element of{0,& mldr;,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 & sdot;2j , j is an element of{0,& mldr;,m-6} , we have a probabilistic algorithm with success probability >= 13-O(2-m/2) for even m >= 6 . Finally, for designed distance 120 & sdot;2(j ), j is an element of{0,& mldr;,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 is an element of{0,& mldr;,& LeftFloor;m/2 & RightFloor;} , 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 <=& LeftFloor;m/2 & RightFloor; , 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 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. 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.