Given a subset K of the unit Euclidean sphere, we estimate the minimal number m = m(K) of hyperplanes that generate a uniform tessellation of K, in the sense that the fraction of the hyperplanes separating any pair x, y ∈ K is nearly proportional to the Euclidean distance between x and y. Random hyperplanes prove to be almost ideal for this problem; they achieve the almost optimal bound m = O(w...

This paper presents a comparative study on the distance optimality for Distance-Preserving Mappings codes generated from different constructions. Few constructions are considered to investigate the reasons behind making codes optimumwhen they reach the upper bound on the sum of the Hamming distances for certain lengths of the permutation sequences. The technique of cyclic-shift prefix or suffix...

This paper is concerned with finite groups G(◦) and G(∗) of order n that are not isomorphic, and where the size of {(u, v) ∈ G×G; u◦v 6= u∗v} is the least possible (with respect to the given n). It surveys the case of 2-groups, discusses the possible generalization of the known results to p-groups, p an odd prime, and establishes the least possible distance in the case when G(∗) is an elementar...

We first present an equivalent definition of relative generalized Hamming weights of a linear code and its subcodes, and we develop a method using finite projective geometry. Making use of the equivalent definition and the projective-geometry method, all of the relative generalized Hamming weights of a 3-dimensional q-ary linear code and its subcodes will be determined.

A counting sequence is a list of all binary words of the same length. Counting sequences of any fixed odd Hamming distance between successive words (codewords) are discussed. Gray codes are examples of counting sequences having a single-bit change between neighboring codewords. We describe some results on Gray codes and highlight some research questions. The spectrum of bit changes or transitio...

Recently, Type IV self-dual codes over rings of order 4 have been introduced as self-dual codes over the rings with the property that all Hamming weights are even. All Type IV self-dual codes over Z4 of lengths up to 16 are known. In this paper, the classi3cation of such codes of length 20 is given. The highest minimum Hamming, Lee and Euclidean weights of Type IV Z4-codes of lengths up to 40 a...

The objectives of this paper are to survey and extend earlier results on the weight distributions of irreducible cyclic codes, present a divisibility theorem and develop bounds on the weights in irreducible cyclic codes. © 2012 Elsevier B.V. All rights reserved.

Repeated root Cyclic and Negacyclic codes over Galois rings have been studied much less than their simple root counterparts. This situation is beginning to change. For example, repeated root codes of length p, where p is the characteristic of the alphabet ring, have been studied under some additional hypotheses. In each one of those cases, the ambient space for the codes has turned out to be a ...

Abstract: In this paper, we study ZpZp[u]-additive codes, where p is prime and u 2 = 0. In particular, we determine a Gray map from ZpZp[u] to Z α+2β p and study generator and parity check matrices for these codes. We prove that a Gray map Φ is a distance preserving map from (ZpZp[u],Gray distance) to (Z α+2β p ,Hamming distance), it is a weight preserving map as well. Furthermore we study the ...

The purpose of this paper is three-fold. Firstly, it aims to demonstrate empirically that networks evolved using different shaped recognition regions in a real-valued shape-space exhibit different dynamics during their formation, and vary in both their capabilities to tolerate antigens and in their memory capacity. Secondly, the paper serves as a useful comparison to previous published work whi...