[Paper Review] A Family of Reversible BCH Codes

Shuxing Li, Cunsheng Ding+1 more|arXiv (Cornell University)|Aug 7, 2016

Coding theory and cryptography18 references16 citations

TL;DR

This paper studies a family of reversible Bose-Chaudhuri-Hocquenghem (BCH) codes, specifically the even-like subcodes of reversible BCH codes, denoted $\overline{{\mathcal{C}}}_{(q,m,\delta)}$. It derives exact dimensions and minimum distances for small $\delta$, establishes bounds for larger $\delta$, and proves that for $\delta = q^\lambda - 1$ with $1 \leq \lambda \leq \lfloor m/2 \rfloor$, the minimum distance is exactly $2\delta$, confirming a conjecture for binary and ternary cases. The codes are shown to achieve optimal or near-optimal parameters in many cases.

ABSTRACT

Cyclic codes are an interesting class of linear codes due to their efficient encoding and decoding algorithms as well as their theoretical importance. BCH codes form a subclass of cyclic codes and are very important in both theory and practice as they have good error-correcting capability and are widely used in communication systems, storage devices and consumer electronics. However, the dimension and minimum distance of BCH codes are not known in general. The objective of this paper is to study the dimension and minimum distance of a family of BCH codes over finite fields, i.e., a class of reversible BCH codes.

Motivation & Objective

• To determine the dimension and minimum distance of a family of reversible BCH codes, specifically the even-like subcodes $\overline{{\mathcal{C}}}_{(q,m,\delta)}$.
• To investigate the parameters of $\overline{{\mathcal{C}}}_{(q,m,\delta)}$ for small values of $\delta$, including $\delta = 2, 3, 4$, under various conditions on $q$ and $m$.
• To establish bounds on the dimension and minimum distance for $\delta = q^\lambda$ with $\frac{m}{2} \leq \lambda \leq m-1$, extending results beyond small $\delta$.
• To explore the optimality of these codes by comparing them with known best cyclic codes, particularly in binary and ternary cases.

Proposed method

• The paper uses the $q$-cyclotomic cosets and coset leaders to analyze the structure of the generator polynomial $\overline{g}_{(q,m,\delta)}(x) = (x-1)\tilde{g}_{(q,m,\delta)}(x)$, where $\tilde{g}_{(q,m,\delta)}(x)$ is the LCM of minimal polynomials for indices $1$ to $\delta-1$ and $n-\delta+1$ to $n-1$.
• It applies the BCH bound to derive lower bounds on the minimum distance, particularly for $\overline{{\mathcal{C}}}_{(q,m,\delta)}$, and uses the sphere packing bound to refine these estimates.
• The dimension is computed via the degree of the generator polynomial, leveraging known results on the number of distinct minimal polynomials in the LCM, especially using Theorem 4 and Corollary 9.
• For specific cases, such as $\delta = 2$, $\delta = 3$, and $\delta = 4$, the paper derives exact parameters using algebraic number theory and properties of finite fields.
• The paper employs numerical experiments and examples to support conjectures, such as $d = 2\delta$ when $\delta = q^\lambda - 1$, and validates them with explicit constructions.
• It compares the parameters of $\overline{{\mathcal{C}}}_{(q,m,\delta)}$ with known tables of best cyclic codes to assess optimality.

Experimental results

Research questions

• RQ1 What is the exact dimension of the even-like subcode $\overline{{\mathcal{C}}}_{(q,m,\delta)}$ for small $\delta$, such as $\delta = 2, 3, 4$?
• RQ2 What is the minimum distance of $\overline{{\mathcal{C}}}_{(q,m,\delta)}$ when $\delta = q^\lambda - 1$ for $1 \leq \lambda \leq \lfloor m/2 \rfloor$?
• RQ3 Under what conditions does $\overline{{\mathcal{C}}}_{(q,m,\delta)}$ achieve the minimum distance $2\delta$, and when is it strictly greater than $\delta$?
• RQ4 How do the parameters of $\overline{{\mathcal{C}}}_{(q,m,\delta)}$ compare to the best known cyclic codes, particularly in binary and ternary cases?
• RQ5 Is the minimum distance of $\overline{{\mathcal{C}}}_{(3,m,4)}$ exactly 8 when $m$ is odd, as conjectured?

Key findings

• For odd $q$ and $m \geq 2$, the code $\overline{{\mathcal{C}}}_{(q,m,2)}$ has parameters $[q^m - 1, q^m - 2 - 2m, 4]$, with minimum distance exactly 4.
• When $q = 2$ and $m \geq 4$, $\overline{{\mathcal{C}}}_{(2,m,3)}$ has parameters $[2^m - 1, 2^m - 2 - 2m, 6]$, achieving minimum distance 6.
• For $q^m \equiv 1 \pmod{3}$ and $m \geq 4$, $\overline{{\mathcal{C}}}_{(q,m,3)}$ has parameters $[q^m - 1, q^m - 2 - 4m, 6]$, with minimum distance 6.
• When $q = 3$ and $m \geq 3$, $\overline{{\mathcal{C}}}_{(3,m,4)}$ has dimension $3^m - 2 - 4m$, and minimum distance is exactly 8 if $m$ is even, and at least 8 if $m$ is odd.
• The conjecture that $\overline{{\mathcal{C}}}_{(q,m,\delta)}$ has minimum distance $2\delta$ when $\delta = q^\lambda - 1$ and $1 \leq \lambda \leq \lfloor m/2 \rfloor$ is supported by numerical evidence and proven for binary and ternary cases.
• The binary reversible BCH code $\overline{{\mathcal{C}}}_{(2,m,\delta)}$ is optimal or among the best known cyclic codes in most tested cases, including $m = 4,5,6$ with $\delta = 3,5,7,9,11,13$.