Minimum distances of BCH codes established using good zero-sets
Minimum distances of primitive narrow-sense BCH codes via good zero-sets
Information Theory
Summary
Determining the exact error-correcting power of BCH codes is difficult, and this paper addresses that challenge. The authors focus on a particular type called primitive narrow-sense BCH codes and find exact minimum distances for several families of them. They introduce a new concept called 'good zero-sets' to characterize when these codes reach their designed error-correcting capability. Using this idea, they develop several methods to construct such zero-sets, enabling them to identify many BCH codes whose error-correcting limits match their expected design.
What this means in practice
- •For communication engineers: Design error-correcting codes with guaranteed minimum distance for reliable data transmission based on new zero-set constructions.
- •For digital storage developers: Implement BCH codes with verified error-correction capabilities to improve data integrity in storage devices.
Authors
Run Zheng
Abstract
Determining the exact minimum distances of BCH codes remains a open problem. We establish the minimum distances of several families of primitive narrow-sense BCH codes, showing that they attain their designed distances. Our approach centers on $\mathbb{F}_q$-good zero-sets, which we introduce through a derivative condition on their vanishing polynomials. We show that a $q$-ary primitive narrow-sense BCH code of length $q^m-1$ and designed distance $2\leqδ\leq q^m-1$ has minimum distance $δ$ if and only if there exists an $\mathbb{F}_q$-good zero-set of cardinality $δ+1$ in the finite field $\mathbb{F}_{q^m}$ with $q^m$ elements. To construct $\mathbb{F}_q$-good zero-sets, we develop several methods based on polynomial substitutions, power maps, and shifted inverses, as well as direct constructions using polynomials of special forms. Together with suitable initial $\mathbb{F}_q$-good zero-sets, including those arising from known minimum-distance results, these methods yield new good zero-sets of various cardinalities and hence families of primitive narrow-sense BCH codes whose minimum distances equal their designed distances. These families cover a broad range of designed distances, with several known minimum-distance results recovered as special cases.