Efficient Algorithms for Constructing Minimum-Weight Codewords in Some Extended Binary BCH Codes
Year: 2024
Type: article
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.
Authors: Amit Berman, Yaron Shany, Itzhak Tamo
Institution: Samsung (Israel)
Language: English
Subfield: Artificial Intelligence
Field: Computer Science
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W4402673736