MacWilliams' Extension Theorem for Bi-Invariant Weights over Finite Principal Ideal Rings

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.Comment: 12 page

Fundus wide subretinal and pigment epithelial abnormalities in macular telangiectasia type 2

Purpose: Macular telangiectasia Type 2 (MacTel) causes glial and photoreceptor cell death in a small, oval patch in the central retina. Beyond this oval area, no disease manifestations have been described so far. Here, we describe a novel pathological aspect of MacTel in the retinal pigment epithelium (RPE) that is not restricted to the clinically affected area but covers the entire retina. Methods: We have studied postmortem eyes from four patients with MacTel by immunohistochemistry and electron microscopy. Results: We found cellular debris in the subretinal space (between photoreceptor outer segments and RPE), consisting mainly of outer segments and RPE components. In healthy eyes, the RPE normally phagocytoses the tips of the continuously growing outer segments, a process considered to be essential for photoreceptor survival. However, in the patients with MacTel, we found no evidence of ongoing outer segment phagocytosis, and the apical surface of the RPE appeared abnormal throughout most of the retina. Conclusion: Reduced outer segment phagocytosis may explain the accumulating debris in the subretinal space but is a surprising finding because visual function in the peripheral retina is normal in patients with MacTel. Nevertheless, the subclinical pathology might induce a specific stress to which the central area is uniquely susceptible

Good Random Matrices over Finite Fields

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.Comment: 25 pages, publishe