[Paper Review] New Families of Triple Error Correcting Codes with BCH Parameters

Carl Bracken|ArXiv.org|Mar 25, 2008

Coding theory and cryptography8 references3 citations

TL;DR

This paper introduces new families of binary triple error-correcting codes with BCH parameters by generalizing the Kasami-Welch theorem. It constructs codes using pairs of functions over $GF(2^n)$, where at least one function is Almost Perfect Nonlinear (APN), and proves that the generalized Fourier transform of these pairs takes only five specific values, ensuring a minimum distance of 7 and matching the weight distribution of the triple error-correcting BCH code.

ABSTRACT

Discovered by Bose, Chaudhuri and Hocquenghem, the BCH family of error correcting codes are one of the most studied families in coding theory. They are also among the best performing codes, particularly when the number of errors being corrected is small relative to the code length. In this article we consider binary codes with minimum distance of 7. We construct new families of codes with these BCH parameters via a generalisation of the Kasami-Welch Theorem.

Motivation & Objective

• To construct new families of binary triple error-correcting codes with BCH parameters using a generalization of the Kasami-Welch theorem.
• To prove that codes constructed from specific pairs of functions over $GF(2^n)$ achieve minimum distance 7.
• To demonstrate that the generalized Fourier transform of the function pairs takes only five specified values, ensuring the desired weight distribution.
• To extend known constructions of triple error-correcting codes by introducing new function pairs with $gcd(n,k)=1$ or odd $n=2t+1$.
• To validate that the dual code has the same weight distribution as the classical triple error-correcting BCH code, confirming its optimality.

Proposed method

• Constructs a parity check matrix $H$ using three mappings $x$, $f(x)$, and $g(x)$ from $GF(2^n)$ to itself, with $f(0)=g(0)=0$, forming a $3n \times (2^n - 1)$ matrix.
• Defines the generalized Fourier transform $F^w(a,b,c) = \sum_{x \in GF(2^n)} (-1)^{\text{Tr}(ax + b f(x) + c g(x))}$ to analyze the weight distribution of the dual code $C^\perp$.
• Uses the MacWilliams identities to compute the weight distribution of $C$ from that of $C^\perp$, relying on the fact that five known weight values in the dual imply identical weight distribution in $C$.
• Applies Lemma 1 on the number of solutions of polynomials over $GF(2^n)$ with $\gcd(k,n)=1$, to bound the number of solutions in subsequent equations.
• Employs iterative substitution and variable transformation (e.g., $u = v w$, $r = w + w^{2^{-k}}$, $z = r p$, $d = q + q^{2^{-k}}$) to reduce high-degree equations to solvable forms.
• Uses the identity $\sum_{x} (-1)^{\text{Tr}(\gamma x)} = 0$ unless $\gamma = 0$, to collapse sums and isolate kernel conditions, ultimately proving that only trivial solutions exist, confirming the transform takes only five values.

Experimental results

Research questions

• RQ1 Can new families of triple error-correcting codes with BCH parameters be constructed using a generalized form of the Kasami-Welch theorem?
• RQ2 Do pairs of functions $\{x^{2^k+1}, x^{2^{tk}+1}\}$ for $t=2,3$ and $\gcd(n,k)=1$ yield codes with minimum distance 7?
• RQ3 Is the generalized Fourier transform $F^w(a,b,c)$ limited to the five values $0, \pm 2^{(n+1)/2}, \pm 2^{(n+3)/2}$ when at least one function is APN?
• RQ4 Can the weight distribution of the dual code be uniquely determined via MacWilliams identities if the transform values are restricted to five known values?
• RQ5 Does the iterative reduction of the equation system via variable substitution lead to a contradiction unless all solutions are trivial, confirming the transform's bounded range?

Key findings

• The paper proves that the generalized Fourier transform $F^w(a,b,c)$ for the function pairs $\{x^{2^k+1}, x^{2^{2k}+1}\}$ and $\{x^{2^k+1}, x^{2^{3k}+1}\}$ takes only the five values $0, \pm 2^{(n+1)/2}, \pm 2^{(n+3)/2}$, which is sufficient to ensure the code has minimum distance 7.
• The construction yields codes with parameters $[2^n - 1, 2^n - 3n - 1, 7]$, matching the triple error-correcting BCH code parameters.
• The dual code $C^\perp$ has the same weight distribution as the dual of the classical triple error-correcting BCH code, confirmed via MacWilliams identities and the five known weight values.
• The proof relies on iterative variable substitution and application of the trace sum identity, ultimately showing that a derived equation has only trivial solutions, leading to a contradiction if non-trivial solutions exist.
• The method confirms that codes constructed from such function pairs are optimal for triple error correction when $\gcd(n,k)=1$ and $n$ is odd.
• The paper establishes that the new families generalize known constructions, including the classic BCH code pair $\{x^3, x^5\}$ and non-BCH codes from prior work, extending their applicability.