An (n, k) group code over a group G is a subset of G which forms a group under componentwise group operation and can be defined in terms of n — k homomorphisms from G to G. In this correspondence, the set of homomorphisms which define Maximum Distance Separable (MDS) group codes defined over cyclic groups are characterized. Each defining homomorphism can be specified by a set of k endomorpbisms...

The support of an [n, k] linear code C over a finite field Fq is the set of all coordinate positions such that at least one codeword has a nonzero entry in each of these coordinate position. The rth generalized Hamming weight dr (C), 1 r k, of C is defined as the minimum of the cardinalities of the supports of all [n, r] subcodes of C. The sequence (d1(C), d2(C), . . . , dk(C)) is called the Ha...

Distances matrices are traditionally analyzed with statistical methods that represent distances as maps such as Metric Multidimensional Scaling (MDS), Generalized Procrustes Analysis (GPA), Individual Differences Scaling (INDSCAL), and DISTATIS. MDS analyzes only one distance matrix at a time while GPA, INDSCAL and DISTATIS extract similarities between several distance matrices. However, none o...

MDS matrices are used as building blocks of diffusion layers in block ciphers, and XOR count is a metric that estimates the hardware implementation cost. In this paper we report the minimum value of XOR counts of 4 × 4 MDS matrices over F24 and F28 , respectively. We give theoretical constructions of Toeplitz MDS matrices and show that they achieve the minimum XOR count. We also prove that Toep...

Recently a lot of attention is paid to the search for efficiently implementable MDS matrices for lightweight symmetric primitives. Most previous work concentrated on locally optimizing the multiplication with single matrix elements. Separate from this line of work, several heuristics were developed to find shortest linear straightline programs. Solving this problem actually corresponds to globa...

A multidimensional file is one whose data are characterized by several attributes, each specified in a given domain. A partial match query on a multidimensional file extracts all data whose attributes match the values of one or more attributes specified in the query. The disk allocation problem of a multidimensional file F on a database system with multiple disks accessible in parallel is the p...

We study multidimensional stochastic equations Xt = x0 + ∫ t 0 B(s,Xs) dWs where x0 is an arbitrary initial state, W is a d-dimensional Wiener process and B : [0, +∞) × IR → IRd2 is a measurable diffusion coefficient. We give sufficient conditions for the existence of weak solutions. Our main result generalizes some results obtained by A. Rozkosz and L. S lomiński [17] and T. Senf [20] for the ...

We consider the motion of a particle in a periodic two dimensional flow perturbed by small (molecular) diffusion. The flow is generated by a divergence free zero mean vector field. The long time behavior corresponds to the behavior of the homogenized process that is diffusion process with the constant diffusion matrix (effective diffusivity). We obtain the asymptotics of the effective diffusivi...

A set of n + k points (k > 0) in projective space of dimension n is said to be an (n + k)-arc if there is no hyperplane containing any n + 1 points of the set. It is well-known that for the projective space PG(n, q), this is equivalent to a maximum distance separable linear code with symbols in the finite field GF(q), of length n + k, dimension n + 1, and distance d = k that satisfies the Singl...