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

Histological Evaluation of Diabetic Neurodegeneration in the Retina of Zucker Diabetic Fatty (ZDF) Rats

In diabetes, retinal dysfunctions exist prior to clinically detectable vasculopathy, however the pathology behind these functional deficits is still not fully established. Previously, our group published a detailed study on the retinal histopathology of type 1 diabetic (T1D) rat model, where specific alterations were detected. Although the majority of human diabetic patients have type 2 diabetes (T2D), similar studies on T2D models are practically absent. To fill this gap, we examined Zucker Diabetic Fatty (ZDF) rats - a model for T2D - by immunohistochemistry at the age of 32 weeks. Glial reactivity was observed in all diabetic specimens, accompanied by an increase in the number of microglia cells. Prominent outer segment degeneration was detectable with changes in cone opsin expression pattern, without a decrease in the number of labelled elements. The immunoreactivity of AII amacrine cells was markedly decreased and changes were detectable in the number and staining of some other amacrine cell subtypes, while most other cells examined did not show any major alterations. Overall, the retinal histology of ZDF rats shows a surprising similarity to T1D rats indicating that despite the different evolution of the disease, the neuroretinal cells affected are the same in both subtypes of diabetes

Computing Symmetrized Weight Enumerators for Lifted Quadratic Residue Codes

The paper describes a method to determine symmetrized weight enumerators of Z p m -linear codes based on the notion of a disjoint weight enumerator. Symmetrized weight enumerators are given for the lifted quadratic residue codes of length 24 modulo 2^m and modulo 3^m, for any positive m

Construction of good LDPC codes using dilation matrices

This paper extends results that have been reported in [1],[6], and [5] for algebraic construction of a low-density matrix to be used as a parity-check matrix for an error-correcting code. After describing the basic idea and the results we finish with a performance diagram that suggests that these codes are as good as random constructions in the area of low SNR and perform better than an algebraic construction tested in [7] in the area of higher SNR. As has been noted in [8], codes with algebraic structure may be easier to implement. The flexibility of the construction we describe may also be advantageous