Orbit Codes over M2(ℱq) and Their Homogeneous Distance

Abstract

Let q be a power of a prime. The lattice of one-sided ideals of the finite unital non-commutative Frobenius ring M2(ℱq) of 2 × 2 matrices over the Galois field (ℱq) is completely analyzed. It turns out that M2(ℱq) 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 M2(ℱq) 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 M2(ℱq) 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 M2(ℱq). 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 M2(ℱq) are constructed via the action of the general linear group GL(2, q) on M2(ℱq) by left translation. The orbit determined by a nonzero nonunit idempotent element of M2(ℱq) forms the nonzero elements of a minimal left ideal of M2(ℱq) 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 M2(ℱq) 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 M2(ℱq) by right translation is used instead.