Relating propelinear and binary G-linear codes

Relating propelinear and binary G-linear codes

In this paper we establish the connections between two di7erent extensions of Z4-linearity for binary Hamming spaces. We present both notions – propelinearity and G-linearity – in the context of isometries and group actions, taking the viewpoint of geometrically uniform codes extended to discrete spaces. We show a double inclusion relation: binary G-linear codes are propelinear codes, and trans...

Ranks of propelinear perfect binary codes

It is proven that for any numbers n = 2 m − 1, m ≥ 4 and r, such that n − log(n + 1) ≤ r ≤ n excluding n = r = 63, n = 127, r ∈ {126, 127} and n = r = 2047 there exists a propelinear perfect binary code of length n and rank r.

Propelinear structure of Z_{2k}-linear codes

Let C be an additive subgroup of Z n 2k for any k ≥ 1. We define a Gray map Φ : Z n 2k −→ Z kn 2 such that Φ(C) is a binary propelinear code and, hence, a Hamming-compatible group code. Moreover, Φ is the unique Gray map such that Φ(C) is Hamming-compatible group code. Using this Gray map we discuss about the nonexistence of 1-perfect binary mixed group code.

Translation-invariant propelinear codes

A class of binary group codes is investigated. These codes are the propelinear codes, deened over the Hamming metric space F n , F = f0; 1g, with a group structure. Generally, they are neither abelian nor translation invariant codes but they have good algebraic and com-binatorial properties. Linear codes and Z 4-linear codes can be seen as a subclass of prope-linear codes. Exactly, it is shown ...

New Binary Linear Codes