On Primitive BCH Codes with Unequal Error Protection Capabilities

Presents a class of binary primitive BCH codes that have unequal-error-protection (UEP) capabilities. The authors use a previous result on the span of their minimum weight vectors to show that binary primitive BCH codes, containing second-order punctured Reed-Muller (RM) codes of the same minimum distance, are binary-cyclic UEP codes. The values of the error correction levels for this class of binary LUEP codes are estimated

Weights in Codes and Genus 2 Curves

We discuss a class of binary cyclic codes and their dual codes. The minimum distance is determined using algebraic geometry, and an application of Weil's theorem. We relate the weights appearing in the dual codes to the number of rational points on a family of genus 2 curves over a finite field

On correlation distribution of Niho-type decimation d=3(pm−1)+1d=3(p^m-1)+1

The cross-correlation problem is a classic problem in sequence design. In this paper we compute the cross-correlation distribution of the Niho-type decimation $d=3(p^m-1)+1$ over $\mathrm{GF}(p^{2m})$ for any prime $p \ge 5$. Previously this problem was solved by Xia et al. only for $p=2$ and $p=3$ in a series of papers. The main difficulty of this problem for $p \ge 5$, as pointed out by Xia et al., is to count the number of codewords of "pure weight" 5 in $p$-ary Zetterberg codes. It turns out this counting problem can be transformed by the MacWilliams identity into counting codewords of weight at most 5 in $p$-ary Melas codes, the most difficult of which is related to a K3 surface well studied in the literature and can be computed. When $p \ge 7$, the theory of elliptic curves over finite fields also plays an important role in the resolution of this problem

The Weight Hierarchies of Linear Codes from Simplicial Complexes

The study of the generalized Hamming weight of linear codes is a significant research topic in coding theory as it conveys the structural information of the codes and determines their performance in various applications. However, determining the generalized Hamming weights of linear codes, especially the weight hierarchy, is generally challenging. In this paper, we investigate the generalized Hamming weights of a class of linear code \C over \bF_q, which is constructed from defining sets. These defining sets are either special simplicial complexes or their complements in \bF_q^m. We determine the complete weight hierarchies of these codes by analyzing the maximum or minimum intersection of certain simplicial complexes and all $r$-dimensional subspaces of \bF_q^m, where 1\leq r\leq {\rm dim}_{\bF_q}(\C)

Kodierungstheorie

[no abstract available

Cryptanalysis of block ciphers and weight divisibility of some binary codes

International audienceThe resistance of an iterated block cipher to most classical attacks can be quantified by some properties of its round function. The involved parameters (nonlinearity, degrees of the derivatives...) for a function F from GF(2^m) into GF(2^m) are related to the weight distribution of a binary linear code C_F of length (2^m − 1) and dimension 2m. In particular, the weight divisibility of C_F appears as an important criterion in the context of linear cryptanalysis and of higher-order differential attacks. When the round function F is a power permutation over GF(2^m), the associated code C_F is the dual of a primitive cyclic code with two zeroes. Therefore, McEliece's theorem provides a powerful tool for evaluating the resistance of some block ciphers to linear and higherorder differential attacks