Recently, a new parameter of a code, referred to as the remoteness, has been introduced.This parameter can be viewed as a dual to the covering radius. It is exactly determined for the cyclic and dihedral groups. In this paper, we consider the group $U_{6n}$ as a subgroup of $S_{2n+3}$ and obtain its remoteness.  We show that the remoteness of the permutation code $U_{6n}$ is $2n+2$.  Moreover, ...

Index 74 Preface The three subjects of the title (codes, matroids, and permutation groups) have many interconnections. In particular, in each case, there is a polynomial which captures a lot of information about the structure: we have the weight enumerator of a code, the Tutte polynomial (or rank polynomial) of a matroid, and the cycle index of a permutation group. With any code is associated a...

| An important problem in the theory and application of block code trellises is to nd a coordinate permutation of a given code to minimize the trellis complexity. In this paper we show that the problem of nding a coordinate permutation that minimizes the number of vertices at a given depth in the minimal trellis for a binary linear block code is NP-complete.

We show that an affine-invariant code C of length p is not permutation equivalent to a cyclic code except in the obvious cases: m = 1 or C is either {0}, the repetition code or its dual. Affine-invariant codes were firstly introduced by Kasami, Lin and Peterson [KLP2] as a generalization of Reed-Muller codes. This class of codes has received the attention of several authors because of its good ...

Permutation coding is a block coding/quantization scheme where the codebook is comprised entirely of permutations of a single starting vector. Permutation codes for the uniform source are developed using a simple algorithm. The performance of these codes is compared against scalar codes and permutation codes developed by different methodologies. It is shown that the algorithm produces codes as ...

Permutation codes are vector quantizers whose codewords are related by permutations and, in one variant, sign changes. Asymptotically, as the vector dimension grows, optimal Variant I permutation code design is identical to optimal entropy-constrained scalar quantizer (ECSQ) design. However, contradicting intuition and previously published assertions, there are finite block length permutation c...

Given an abstract group G, an N dimensional orthogonal matrix representation G of G, and an “initial vector” x ∈ R , Slepian defined the group code generated by the representation G to be the set of vectors Gx. If G is a group of permutation matrices, the set Gx is called a “permutation code”. For permutation codes a ’stack algorithm’ decoder exists that, in the presence of low noise, produces ...