Orbit Codes over M 2(ℱ q ) and Their Homogeneous Distance
E C T Corro, John Mark Lampos, Herbert S. Palines, Virgilio P. Sıson · Journal of Physics Conference Series · 2018
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.