Efficient Algorithms for Constructing Minimum-Weight Codewords in Some Extended Binary BCH Codes
Amit Berman, Yaron Shany, Itzhak Tamo
TL;DR
This work develops explicit, low-complexity methods for constructing minimum-weight codewords in extended binary BCH codes of length $n=2^m$, focusing on designed distances $d(m,s,i)=2^{m-1-s}-2^{m-1-i-s}$. By reframing the problem as solving a system of $i-1$ multilinear equations in $2i$ variables and leveraging the down-conversion/up-conversion theorems, the authors obtain $O(m^3)$ algorithms that yield minimum-weight supports in the form $X+\mathrm{Span}_{\mathbb{F}_2}(B)$, with fixed $|X|$ and $|B|$. They provide deterministic and probabilistic constructions for specific cases: $i=2$ (deterministic for even $m$, probabilistic for odd $m$), $i=3$ (even $m$ with success probability at least $1/3-O(2^{-m/2})$), and $i=4$ (deterministic for $m$ divisible by $4$), as well as a Gold-function-based method when $2i|m$ that extends to multiple designed distances via up-conversion. The results generalize and extend prior explicit constructions (e.g., GK12) and offer practical tools for debugging and testing decoders by supplying concrete minimum-weight codewords, while also re-proving the conversion theorems in a novel, evaluated-polynomial framework. Open questions remain about universal independent-coordinate solutions for all parameter regimes and the potential use of Gröbner-basis methods or higher-degree affine generators. Overall, the paper delivers constructive, scalable techniques for a longstanding problem in BCH code theory with implications for both theory and applications in coding and testing of decoders.
Abstract
We present $O(m^3)$ algorithms for specifying the support of minimum-weight words 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. 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 $6\cdot 2^j$, we have a deterministic algorithm for even $m\geq 4$, and a probabilistic algorithm with success probability $1-O(2^{-m})$ for odd $m>4$. For designed distance $28\cdot 2^j$, we have a probabilistic algorithm with success probability $\geq 1/3-O(2^{-m/2})$ for even $m\geq 6$. Finally, for designed distance $120\cdot 2^j$, we have a deterministic algorithm for $m\geq 8$ divisible by $4$. We also present a construction via Gold functions when $2i|m$. Our construction builds on results of Kasami and Lin (IEEE T-IT, 1972), who proved that for extended binary BCH codes of designed distance $d(m,s,i)$, the minimum distance equals the designed distance. Their proof makes use of a non-constructive result of Berlekamp (Inform. Contrl., 1970), 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 argument of Berlekamp by a low-complexity algorithm. In one aspect, we extends the results of Grigorescu and Kaufman (IEEE T-IT, 2012), who presented explicit minimum-weight words for designed distance $6$ (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. However, the minimum-weight words we construct are not affine generators for designed distance $>6$.