Explicit constructions of quasi-uniform codes from groups

Abstract

We address the question of constructing explicitly quasi-uniform codes from groups. We determine the size of the codebook, the alphabet and the minimum distance as a function of the corresponding group, both for abelian and some nonabelian groups. Potentials applications comprise the design of almost affine codes and non-linear network codes.

Almost affine locally repairable codes and matroid theory

“…Cyclic flats seem to deal effectively with the parameters (n, k, d, r) and seem to be a very useful tool on the problems we are dealing with. By the use of a construction of quasi-uniform codes in [15], we give a construction of optimal LRCs that are almost affine and vector-linear, have high rate, and have alphabet size 4 for any n = 4i + 4 when i ≥ 1.…”

“…Quasi-uniform codes were introduced in [14], and include almost affine codes as a special case. An explicit construction of quasi-uniform codes from groups is given in [15]. This construction can be characterized as follows.…”

“…The minimum distance d of C and the size of its projections was given in [15] as follows. For X ∈ [n], let G X = ∩ i∈X G i , then…”

Applications of quasi-uniform codes to storage

“…Since quasi-uniform distributions may be obtained from finite groups [3], finite groups in turn give rise to quasiuniform codes [8]. Let us shortly recall these constructions.…”

“…Quasi-uniform codes are defined with respect to an underlying probability distribution which is quasi-uniform [1], as will be explained in Section II, and quasi-uniform distributions (and in turn quasi-uniform codes [2]) can be constructed from finite groups and their subgroups [3]. When quasi-uniform codes come from finite groups, the underlying group structure can be exploited to derive code properties, such as the minimum distance [8].…”