arXiv subjects
Publications and source records attributed to Thomas Honold.
The results of [1,2] on linear homogeneous two-weight codes over finite Frobenius rings are exended in two ways: It is shown that certain non-projective two-weight codes give rise to strongly regular graphs in the way described in [1,2]. Secondly, these codes are used to define a dual two-weight code and strongly regular graph similar to the classical case of projective linear two-weight codes over finite fields [3].
The results of [1,2] on linear homogeneous two-weight codes over finite Frobenius rings are exended in two ways: It is shown that certain non-projective two-weight codes give rise to strongly regular graphs in the way described in [1,2]. Secondly, these codes are used to define a dual two-weight code and strongly regular graph similar to the classical case of projective linear two-weight codes over finite fields [3].
A finite ring R and a weight w on R satisfy the Extension Property if every R-linear w-isometry between two R-linear codes in R^n extends to a monomial transformation of R^n that preserves w. MacWilliams proved that finite fields with the Hamming weight satisfy the Extension Property. It is known that finite Frobenius rings with either the Hamming weight or the homogeneous weight satisfy the Extension Property. Conversely, if a finite ring with the Hamming or homogeneous weight satisfies the Extension Property, then the ring is Frobenius. This paper addresses the question of a characterization of all bi-invariant weights on a finite ring that satisfy the Extension Property. Having solved this question in previous papers for all direct products of finite chain rings and for matrix rings, we have now arrived at a characterization of these weights for finite principal ideal rings, which form a large subclass of the finite Frobenius rings. We do not assume commutativity of the rings in question.
The random matrix uniformly distributed over the set of all m-by-n matrices over a finite field plays an important role in many branches of information theory. In this paper a generalization of this random matrix, called k-good random matrices, is studied. It is shown that a k-good random m-by-n matrix with a distribution of minimum support size is uniformly distributed over a maximum-rank-distance (MRD) code of minimum rank distance min{m,n}-k+1, and vice versa. Further examples of k-good random matrices are derived from homogeneous weights on matrix modules. Several applications of k-good random matrices are given, establishing links with some well-known combinatorial problems. Finally, the related combinatorial concept of a k-dense set of m-by-n matrices is studied, identifying such sets as blocking sets with respect to (m-k)-dimensional flats in a certain m-by-n matrix geometry and determining their minimum size in special cases.
Linear operator broadcast channel (LOBC) models the scenario of multi-rate packet broadcasting over a network, when random network coding is applied. This paper presents the framework of algebraic coding for LOBCs and provides a Hamming-like upper bound on (multishot) subspace codes for LOBCs.
The average weight distribution of a regular low-density parity-check (LDPC) code ensemble over a finite field is thoroughly analyzed. In particular, a precise asymptotic approximation of the average weight distribution is derived for the small-weight case, and a series of fundamental qualitative properties of the asymptotic growth rate of the average weight distribution are proved. Based on this analysis, a general result, including all previous results as special cases, is established for the minimum distance of individual codes in a regular LDPC code ensemble.
Recent research indicates that packet transmission employing random linear network coding can be regarded as transmitting subspaces over a linear operator channel (LOC). In this paper we propose the framework of linear operator broadcast channels (LOBCs) to model packet broadcasting over LOCs, and we do initial work on the capacity region of constant-dimension multiplicative LOBCs(CMLOBCs), a generalization of broadcast erasure channels. Two fundamental problems regarding CMLOBCs are addressed-finding necessary and sufficient conditions for degradation and deciding whether time sharing suffices to achieve the boundary of the capacity region in the degraded case.
The problem of finding good linear codes for joint source-channel coding (JSCC) is investigated in this paper. By the code-spectrum approach, it has been proved in the authors' previous paper that a good linear code for the authors' JSCC scheme is a code with a good joint spectrum, so the main task in this paper is to construct linear codes with good joint spectra. First, the code-spectrum approach is developed further to facilitate the calculation of spectra. Second, some general principles for constructing good linear codes are presented. Finally, we propose an explicit construction of linear codes with good joint spectra based on low density parity check (LDPC) codes and low density generator matrix (LDGM) codes.