New cubic self-dual codes of length 54, 60 and 66

We study the construction of quasi-cyclic self-dual codes, especially of binary cubic ones. We consider the binary quasi-cyclic codes of length 3\ell with the algebraic approach of [9]. In particular, we improve the previous results by constructing 1 new binary [54, 27, 10], 6 new [60, 30, 12] and 50 new [66, 33, 12] cubic self-dual codes. We conjecture that there exist no more binary cubic self-dual codes with length 54, 60 and 66.Comment: 8 page

The 2-distance coloring of the Cartesian product of cycles using optimal Lee codes

AbstractLet Cm be the cycle of length m. We denote the Cartesian product of n copies of Cm by G(n,m):=Cm□Cm□⋯□Cm. The k-distance chromatic number χk(G) of a graph G is χ(Gk) where Gk is the kth power of the graph G=(V,E) in which two distinct vertices are adjacent in Gk if and only if their distance in G is at most k. The k-distance chromatic number of G(n,m) is related to optimal codes over the ring of integers modulo m with minimum Lee distance k+1. In this paper, we consider χ2(G(n,m)) for n=3 and m≥3. In particular, we compute exact values of χ2(G(3,m)) for 3≤m≤8 and m=4k, and upper bounds for m=3k or m=5k, for any positive integer k. We also show that the maximal size of a code in Z63 with minimum Lee distance 3 is 26

A new class of codes for Boolean masking of cryptographic computations

We introduce a new class of rate one-half binary codes: {\bf complementary information set codes.} A binary linear code of length $2n$ and dimension $n$ is called a complementary information set code (CIS code for short) if it has two disjoint information sets. This class of codes contains self-dual codes as a subclass. It is connected to graph correlation immune Boolean functions of use in the security of hardware implementations of cryptographic primitives. Such codes permit to improve the cost of masking cryptographic algorithms against side channel attacks. In this paper we investigate this new class of codes: we give optimal or best known CIS codes of length $<132.$ We derive general constructions based on cyclic codes and on double circulant codes. We derive a Varshamov-Gilbert bound for long CIS codes, and show that they can all be classified in small lengths $\le 12$ by the building up construction. Some nonlinear permutations are constructed by using $\Z_4$-codes, based on the notion of dual distance of an unrestricted code.Comment: 19 pages. IEEE Trans. on Information Theory, to appea

Construction of quasi-cyclic self-dual codes

There is a one-to-one correspondence between $\ell$-quasi-cyclic codes over a finite field $\mathbb F_q$ and linear codes over a ring $R = \mathbb F_q[Y]/(Y^m-1)$. Using this correspondence, we prove that every $\ell$-quasi-cyclic self-dual code of length $m\ell$ over a finite field $\mathbb F_q$ can be obtained by the {\it building-up} construction, provided that char $(\mathbb F_q)=2$ or $q \equiv 1 \pmod 4$, $m$ is a prime $p$, and $q$ is a primitive element of $\mathbb F_p$. We determine possible weight enumerators of a binary $\ell$-quasi-cyclic self-dual code of length $p\ell$ (with $p$ a prime) in terms of divisibility by $p$. We improve the result of [3] by constructing new binary cubic (i.e., $\ell$-quasi-cyclic codes of length $3\ell$) optimal self-dual codes of lengths $30, 36, 42, 48$ (Type I), 54 and 66. We also find quasi-cyclic optimal self-dual codes of lengths 40, 50, and 60. When $m=5$, we obtain a new 8-quasi-cyclic self-dual $[40, 20, 12]$ code over $\mathbb F_3$ and a new 6-quasi-cyclic self-dual $[30, 15, 10]$ code over $\mathbb F_4$. When $m=7$, we find a new 4-quasi-cyclic self-dual $[28, 14, 9]$ code over $\mathbb F_4$ and a new 6-quasi-cyclic self-dual $[42,21,12]$ code over $\mathbb F_4$.Comment: 25 pages, 2 tables; Finite Fields and Their Applications, 201

Identifying Codes in q-ary Hypercubes

Let q be any integer ≥ 2. In this paper, we consider the q-ary n-dimensional cube whose vertex set is Z n q and two vertices (x1,..., xn) and (y1,..., yn) are adjacent if their Lee distance is 1. As a natural extension of identifying codes in binary Hamming spaces, we further study identifying codes in the above q-ary hypercube. We let M q t (n) denote the smallest cardinality of t-identifying codes of length n in Z n q. Little is known about ternary or quaternary identifying codes. It is known [2, 14] that M 2 1(n)