On M.D.S. codes, arcs in PG(n,q) with q even, and a solution of three fundamental problems of B. Segre.
On mixed codes with covering radius 1 and minimum distance 2.
On Multiple Deletion Codes
In 1965 Levenshtein introduced the deletion correcting codes and found an asymptotically optimal family of 1-deletion correcting codes. During the years there has been a little or no research on t-deletion correcting codes for larger values of t. In this paper, we consider the problem of finding the maximal cardinality L2(n;t) of a binary t-deletion correcting code of length n. We construct an infinite family of binary t-deletion correcting codes. By computer search, we construct t-deletion codes...
On optimal linear codes over .
On perfect-like binary and non-binary linear codes - a brief survey.
On sets of vectors of a finite vector space in which every subset of basis size is a basis
It is shown that the maximum size of a set of vectors of a -dimensional vector space over , with the property that every subset of size is a basis, is at most , if , and at most , if , where and is prime. Moreover, for , the sets of maximum size are classified, generalising Beniamino Segre’s “arc is a conic” theorem. These results have various implications. One such implication is that a matrix, with and entries from , has columns which are linearly dependent. Another is...
On some Optimal (n,t,1,2) and (n,t,1,3) Super Imposed Codes
Partially supported by the Technical University of Gabrovo under Grant C-801/2008One of the main problems in the theory of superimposed codes is to find the minimum length N for which an (N, T,w, r) superimposed code exists for given values of T , w and r. Let N(T,w, r) be the minimum length N for which an (N, T,w, r) superimposed code exists. The (N, T,w, r) superimposed code is called optimal when N = N(T,w, r). The values of N(T, 1, 2) are known for T ≤ 12 and the values of N(T, 1, 3) are known for...
On some subsemigroups of a free semigroup.
On the Automorphism Groups of some AG-Codes Based on Ca;b Curves
*Partially supported by NATO.We study Ca,b curves and their applications to coding theory. Recently, Joyner and Ksir have suggested a decoding algorithm based on the automorphisms of the code. We show how Ca;b curves can be used to construct MDS codes and focus on some Ca;b curves with extra automorphisms, namely y^3 = x^4 + 1, y^3 = x^4 - x, y^3 - y = x^4. The automorphism groups of such codes are determined in most characteristics.
On the classification of 3-dimensional non-associative division algebras over -adic fields
Let be a prime and a -adic field (a finite extension of the field of -adic numbers ). We employ the main results in [12] and the arithmetic of elliptic curves over to reduce the problem of classifying 3-dimensional non-associative division algebras (up to isotopy) over to the classification of ternary cubic forms over (up to equivalence) with no non-trivial zeros over . We give an explicit solution to the latter problem, which we then relate to the reduction type of the jacobian...
On the coding theorem for decomposable discrete information channels. I
On the coding theorem for decomposable discrete information channels. II
On the Construction of Codes from an Asymptotically Good Tower over F8
In 2002, van der Geer and van der Vlugt gave explicit equations for an asymptotically good tower of curves over the field F8. In this paper, we will present a method for constructing Goppa codes from these curves as well as explicit constructions for the third level of the tower. The approach is to find an associated plane curve for each curve in the tower and then to use the algorithms of Haché and Le Brigand to find the corresponding Goppa codes.
On the construction of dense lattices with a given automorphisms group
We consider the problem of constructing dense lattices in with a given non trivial automorphisms group. We exhibit a family of such lattices of density at least , which matches, up to a multiplicative constant, the best known density of a lattice packing. For an infinite sequence of dimensions , we exhibit a finite set of lattices that come with an automorphisms group of size , and a constant proportion of which achieves the aforementioned lower bound on the largest packing density. The algorithmic...
On the Error-Correcting Performance of some Binary and Ternary Linear Codes
In this work, we determine the coset weight spectra of all binary cyclic codes of lengths up to 33, ternary cyclic and negacyclic codes of lengths up to 20 and of some binary linear codes of lengths up to 33 which are distance-optimal, by using some of the algebraic properties of the codes and a computer assisted search. Having these weight spectra the monotony of the function of the undetected error probability after t-error correction P(t)ue (C,p) could be checked with any precision for a linear...
On the failing cases of the Johnson bound for error-correcting codes.
On the minimum distance of ideals in group algebras
On the most weight vectors in a dimension binary code.
On the Non-Existence of Certain M.D.S. Codes and Projective Planes.
On the number of zero trace elements in polynomial bases for F2n.
Let Fq denote the finite field of q elements. O. Ahmadi and A. Menezes have recently considered the question about the possible number of elements with zero trace in polynomial bases of F2n over F2. Here we show that the Weil bound implies that there is such a basis with n + O(log n) zero-trace elements.