Orbit Codes over M 2(q) and Their Homogeneous Distance

Abstract

© Published under licence by IOP Publishing Ltd. Let q be a power of a prime. The lattice of one-sided ideals of the finite unital non-commutative Frobenius ring M 2(q) of 2 × 2 matrices over the Galois field (q) is completely analyzed. It turns out that M 2(q) is a principal left semi-local ring in which each left ideal is generated by an idempotent element. The explicit forms of the non-trivial idempotents of M 2(q) are determined to give q + 1 proper non-trivial left maximal ideals each with q elements. These are exactly the minimal left ideals as well. Using the structure of M 2(q) as a partial ordering of ideals, the generalized Möbius and Euler phi functions are applied to derive the explicit form of the homogeneous weight function on M 2(q). This weight depends on whether the element is the zero element, a zero divisor or a unit. A zero divisor gives the largest homogeneous weight. Moreover, orbit codes over M 2(q) are constructed via the action of the general linear group GL(2, q) on M 2(q) by left translation. The orbit determined by a nonzero nonunit idempotent element of M 2(q) forms the nonzero elements of a minimal left ideal of M 2(q) which are all zero divisors. Consequently, it is shown that the minimum homogeneous distance of the orbit code generated by a nonzero nonunit idempotent element of M 2(q) approaches the Plotkin upper bound as the field size q becomes larger. Analogous results are obtained when the lattice of right ideals is considered and the action of GL(2, q) on M 2(q) by right translation is used instead.

Source or Periodical Title

Journal of Physics: Conference Series

ISSN

17426588

Document Type

Article

This document is currently not available here.

Share

COinS