Difference systems of sets can be used to transform an arbitrary linear code to a coset of a linear code with a given comma-free index by means of a minimal increase of its length. The paper discusses some constructions of difference systems of sets obtained from cyclic difference sets and finite geometry.

Linear codes with complementary duals (abbreviated LCD) are linear codes whose intersection with their dual are trivial. When they are binary, they play an important role in armoring implementations against side-channel attacks and fault injection attacks. Non-binary LCD codes in characteristic 2 can be transformed into binary LCD codes by expansion. In this paper, we introduce a general constr...

Cyclic codes are an important class of linear codes, whose weight distribution have been extensively studied. So far, most of previous results obtained were for cyclic codes with no more than three zeros. Recently, [37] constructed a class of cyclic codes with arbitrary number of zeros, and computed the weight distributions for several cases. In this paper, we determine the weight distribution ...

In this paper we describe a method of finding differential patterns that may be used to attack reduced versions of SHA-1. We show that the problem of finding optimal differential patterns for SHA-1 is equivalent to the problem of finding minimal weight codeword in a linear code. Finally, we present a number of patterns of different lengths suitable for finding collisions and near-collisions and...

Let Ck(n, q) be the p-ary linear code defined by the incidence matrix of points and k-spaces in PG(n, q), q = p, p prime, h ≥ 1. In this paper, we show that there are no codewords of weight in the open interval ] q −1 q−1 , 2q[ in Ck(n, q) \ Cn−k(n, q) ⊥ which implies that there are no codewords with this weight in Ck(n, q) \ Ck(n, q) ⊥ if k ≥ n/2. In particular, for the code Cn−1(n, q) of poin...

one of the most important problems of coding theory is to construct codes with beat possible minimum distances. In this paper, we use the algebraic structure of quasi-cyclic codes and the BCH type bound on the minimum distance to search for quasi-cyclic codes over IFS, the finite field with five elements, which improve the minimum distances of best-known linear codes. We construct 15 new linear...

We consider linear error correcting codes associated to higher-dimensional projective varieties defined over a finite field. The problem of determining the basic parameters of such codes often leads to some interesting and difficult questions in combinatorics and algebraic geometry. This is illustrated by codes associated to Schubert varieties in Grassmannians, called Schubert codes, which have...

For a linear code C of length n and dimension k, Wolf noticed that the trellis state complexity s(C) of C is upper bounded by w(C) := min(k, n− k). In this paper we point out some new lower bounds for s(C). In particular, if C is an Algebraic Geometric code, then s(C) ≥ w(C) − (g − a), where g is the genus of the underlying curve and a is the abundance of the code.

With the help of a computer, we obtain the minimum distance of some codes belonging to two families of Z2k -linear codes: the first is the generalized Kerdock codes which aren’t as good as the best linear codes and the second is the Hensel lift of quadratic residue codes. In the latter, we found new codes with same minimum distances as the best linear codes of same length and same cardinality. ...