The covering radius of long primitive ternary BCH codes

This thesis is about the generalisation of a method to determine an asymptotic upper bound for the covering radius of primitive BCH codes. The method was introduced by S. D. Cohen in the mid-1990s for binary codes. It reduces the coding-theoretical problem to the complete splitting of a single polynomial F(x) over a finite field, which is then established using results that have their roots in ramification theory of function fields. The opening chapter introduces the covering radius problem for BCH codes along with its full coding-theoretical background and some history. As a first result, the transformation from the covering radius problem to a polynomial splitting problem is extended to primitive p-ary BCH codes, where p is an arbitrary prime. The process, during which an explicit "ready-to-use" form of the general F is derived, is summarised in one theorem (Theorem 6).The foundations for arranging the splitting of F (via certain adjustable coefficients) were laid in previous work by Cohen, which is presented in extracts. By combining the key strategy of this with new ideas to meet the special requirements of the non-binary case, sufficient criteria for the splitting are obtained; these come in the form of conditions on polynomials ƒ[0] and ƒ[1], where F has been parameterised as ƒ[0] + uƒ[1] (u an indeterminate). Several other lemmas are proved to deal with the establishing of the conditions. All these results are valid for arbitrary primes p ≥ 3, so that with this the desired general version of the method has been made available

Several families of ternary negacyclic codes and their duals

Constacyclic codes contain cyclic codes as a subclass and have nice algebraic structures. Constacyclic codes have theoretical importance, as they are connected to a number of areas of mathematics and outperform cyclic codes in several aspects. Negacyclic codes are a subclass of constacyclic codes and are distance-optimal in many cases. However, compared with the extensive study of cyclic codes, negacyclic codes are much less studied. In this paper, several families of ternary negacyclic codes and their duals are constructed and analysed. These families of negacyclic codes and their duals contain distance-optimal codes and have very good parameters in general

Advances in Syndrome Coding based on Stochastic and Deterministic Matrices for Steganography

Steganographie ist die Kunst der vertraulichen Kommunikation. Anders als in der Kryptographie, wo der Austausch vertraulicher Daten für Dritte offensichtlich ist, werden die vertraulichen Daten in einem steganographischen System in andere, unauffällige Coverdaten (z.B. Bilder) eingebettet und so an den Empfänger übertragen. Ziel eines steganographischen Algorithmus ist es, die Coverdaten nur geringfügig zu ändern, um deren statistische Merkmale zu erhalten, und möglichst in unauffälligen Teilen des Covers einzubetten. Um dieses Ziel zu erreichen, werden verschiedene Ansätze der so genannten minimum-embedding-impact Steganographie basierend auf Syndromkodierung vorgestellt. Es wird dabei zwischen Ansätzen basierend auf stochastischen und auf deterministischen Matrizen unterschieden. Anschließend werden die Algorithmen bewertet, um Vorteile der Anwendung von Syndromkodierung herauszustellen

On Saturating Sets in Small Projective Geometries

AbstractA set of points, S⊆PG(r, q), is said to be ϱ -saturating if, for any point x∈PG(r, q), there exist ϱ+ 1 points in S that generate a subspace in which x lies. The cardinality of a smallest possible set S with this property is denoted by k(r, q,ϱ ). We give a short survey of what is known about k(r, q, 1) and present new results for k(r, q, 2) for small values of r and q. One construction presented proves that k(5, q, 2) ≤ 3 q+ 1 forq= 2, q≥ 4. We further give an upper bound onk (ϱ+ 1, pm, ϱ)

Covering Radius 1985-1994

We survey important developments in the theory of covering radius during the period 1985-1994. We present lower bounds, constructions and upper bounds, the linear and nonlinear cases, density and asymptotic results, normality, specific classes of codes, covering radius and dual distance, tables, and open problems