Harald Fripertinger

Error-correcting linear codes : classification by isometry and applications

Linear Codes.- Bounds and Modifications.- Finite Fields.- Cyclic Codes.- Mathematics and Audio Compact Discs.- Enumeration of Isometry Classes.- Solving Systems of Diophantine Linear Equations.- Linear Codes with a Prescribed Minimum Distance.- The General Case.

Cycle indices of linear, affine, and projective groups

Abstract The Polya cycle indices for the natural actions of the general linear groups and affine groups (on a vector space) and for the projective linear groups (on a projective space) over a finite field are computed. Finally it is demonstrated how to enumerate isometry classes of linear codes by using these cycle indices.

Isometry Classes of Indecomposable Linear Codes

In the constructive theory of linear codes, we can restrict attention to the isometry classes of indecomposable codes, as it was shown by Slepian. We describe these classes as orbits and we demonstrate how they can be enumerated using cycle index polynomials. The necessary tools are already incorporated in SYMMETRICA, a (public domain) computer algebra package devoted to representation theory and combinatorics of symmetric groups and of related classes of groups. Moreover, we describe how systems of representatives of these classes can be evaluated using double coset methods.

Tone rows and tropes

Applying various methods based on group actions we provide a complete classification of tone rows in the twelve-tone scale. The main objects of the present paper are the orbits of tone rows under the action of the direct product of two dihedral groups. This means that tone rows are considered to be equivalent if and only if they can be constructed by transposition, inversion, retrograde, and/or time shift (rotation) from a single row. We determine the orbit, the normal form, the stabilizer class of a tone row, its trope structure, diameter distance, and chord diagram. A database provides complete information on all pairwise non-equivalent tone rows. It can be accessed via http://www.uni-graz.at/∼fripert/db/. Bigger orbits of tone rows are studied when we allow further operations on tone rows such as the quart-circle (multiplication), the five-step (multiplication in the time domain), or the interchange of parameters.

On Covariant Embeddings of a Linear Functional Equation with Respect to an Analytic Iteration Group

Let a(x), b(x), p(x) be formal power series in the indeterminate x over (i.e. elements of the ring of such series) such that ord a(x) = 0, ord p(x) = 1 and p(x) is embeddable into an analytic iteration group in . By a covariant embedding of the linear functional equation (for the unknown series ) with respect to . In this paper we solve the system ((Co1), (Co2)) (of so-called cocycle equations) completely, describe when and how the boundary conditions (B1) and (B2) can be satisfied, and present a large class of equations (L) together with iteration groups for which there exist covariant embeddings of (L) with respect to .

Enumeration of mosaics

Mosaics are orbits of partitions arising from music theoretical investigations. Various theorems from the field of “enumeration under finite group actions” are applied for enumerating mosaics. In other words, it is demonstrated how to enumerateG-orbits of partitions of given size, block-type or stabilizer-type.

The number of invariant subspaces under a linear operator on finite vector spaces

Let

ON THE FORMAL FIRST COCYCLE EQUATION FOR ITERATION GROUPS OF TYPE II

V

Enumeration, Construction and Random Generation of Block Codes

be an

Peter Lackner/Tropical Investigations

n