[Paper Review] A Triple-Error-Correcting Cyclic Code from the Gold and Kasami-Welch APN Power Functions

Xiangyong Zeng, Jinyong Shan+1 more|arXiv (Cornell University)|Mar 31, 2010

Coding theory and cryptography19 references4 citations

TL;DR

This paper constructs a new triple-error-correcting binary cyclic code $ ilde{ m C}_{1,3,13}$ of length $2^m - 1$ for odd $m /geq 5$, defined by zeros $eta$, $eta^3$, and $eta^{13}$, where $eta$ is a primitive element in $ ext{GF}(2^m)$. The code is derived from the Gold ($x^3$) and Kasami-Welch ($x^{13}$) almost perfect nonlinear (APN) power functions, and it is proven to have the same weight distribution as the classical primitive BCH code $ ilde{ m C}_{1,3,5}$, establishing its optimality in error-correcting capability and spectral properties.

ABSTRACT

Based on a sufficient condition proposed by Hollmann and Xiang for constructing triple-error-correcting codes, the minimum distance of a binary cyclic code $\mathcal{C}_{1,3,13}$ with three zeros $α$, $α^3$, and $α^{13}$ of length $2^m-1$ and the weight divisibility of its dual code are studied, where $m\geq 5$ is odd and $α$ is a primitive element of the finite field $\mathbb{F}_{2^m}$. The code $\mathcal{C}_{1,3,13}$ is proven to have the same weight distribution as the binary triple-error-correcting primitive BCH code $\mathcal{C}_{1,3,5}$ of the same length.

Motivation & Objective

• To construct a new triple-error-correcting binary cyclic code of length $2^m - 1$ for odd $m \geq 5$.
• To identify code parameters derived from the Gold and Kasami-Welch APN power functions $x^3$ and $x^{13}$.
• To prove that the code $\mathcal{C}_{1,3,13}$ has the same weight distribution as the well-known primitive BCH code $\mathcal{C}_{1,3,5}$.
• To verify that the dual code of $\mathcal{C}_{1,3,13}$ has weights divisible by $2^{(m-1)/2}$, satisfying Hollmann and Xiang's sufficient condition for weight distribution equivalence.
• To extend the known class of codes with identical weight distributions to include a new pair of APN exponents derived from different APN function families.

Proposed method

• Leveraging Hollmann and Xiang's sufficient condition: if a binary cyclic code of length $2^m - 1$ has minimum distance $\geq 7$ and dual code weights divisible by $2^{(m-1)/2}$, then it shares the same weight distribution as $\mathcal{C}_{1,3,5}$.
• Analyzing the dual code $\mathcal{C}_{1,3,13}^\perp$ as the set of vectors $\left(\text{Tr}^m_1(\epsilon x + \gamma x^3 + \delta x^{13})\right)_{x \in \mathbb{F}_{2^m}^*}$ for $\epsilon, \gamma, \delta \in \mathbb{F}_{2^m}$.
• Using the MacWilliams identity to equate weight distributions of $\mathcal{C}_{1,3,13}$ and $\mathcal{C}_{1,3,5}$ by comparing the weight distributions of their duals.
• Performing exhaustive computational experiments for $m = 5, 7, 9, 11$ to test all known APN exponent pairs, identifying $(d_1, d_2) = (3,13)$ as a new valid pair.
• Applying algebraic techniques involving the $2$-adic valuation of coefficients and cyclotomic equivalence to bound the maximum weight of certain linear combinations.
• Proving that $M(m;3,13) = (m - \gcd(m,r))/2$ using a generalized version of a result from [17], confirming the maximum possible weight of codewords in the dual code.

Experimental results

Research questions

• RQ1 Does the binary cyclic code $\mathcal{C}_{1,3,13}$ with zeros $\alpha, \alpha^3, \alpha^{13}$ have minimum distance at least 7?
• RQ2 Does the dual code $\mathcal{C}_{1,3,13}^\perp$ have all weights divisible by $2^{(m-1)/2}$ for odd $m \geq 5$?
• RQ3 Does $\mathcal{C}_{1,3,13}$ have the same weight distribution as the primitive BCH code $\mathcal{C}_{1,3,5}$?
• RQ4 Can the pair $(3,13)$, derived from the Gold and Kasami-Welch APN functions, generate a new triple-error-correcting code with the same weight distribution as $\mathcal{C}_{1,3,5}$?
• RQ5 Are there other APN exponent pairs beyond known constructions that yield codes with identical weight distributions to $\mathcal{C}_{1,3,5}$?

Key findings

• The code $\mathcal{C}_{1,3,13}$ is proven to have minimum distance exactly 7, satisfying the triple-error-correcting requirement.
• The dual code $\mathcal{C}_{1,3,13}^\perp$ has all weights divisible by $2^{(m-1)/2}$, fulfilling the key condition from Hollmann and Xiang's criterion.
• The weight distribution of $\mathcal{C}_{1,3,13}$ is identical to that of the classical primitive BCH code $\mathcal{C}_{1,3,5}$, as confirmed via MacWilliams identity and computational verification for $m = 5, 7, 9, 11$.
• The pair $(d_1, d_2) = (3,13)$ is a new valid pair not explainable by prior constructions such as $\{2^r+1, 2^{3r}+1\}$ or $\{2^{(m+1)/2}+1, (2^{(m+1)/2}+1)^2\}$, and is marked with $\bigstar$ in Table 3 as newly discovered.
• The maximum weight $M(m;3,13)$ of linear combinations of the form $\text{Tr}^m_1(\epsilon x + \gamma x^3 + \delta x^{13})$ is shown to be $k = (m - \gcd(m,r))/2$, matching the bound for $M(m;3)$, confirming tightness of the bound.
• The construction generalizes to other APN functions: if $f(x)$ and $g(x)$ are different APN functions over $\mathbb{F}_{2^m}$, then the code with dual $\mathcal{C}^\perp = \{ \text{Tr}^m_1(\epsilon x + \gamma f(x) + \delta g(x)) \}$ may yield new codes with identical weight distributions under similar conditions.