Primitive idempotents in central simple algebras over $\mathbb{F}_q(t)$ with an application to coding theory

Abstract

We consider the algorithmic problem of computing a primitive idempotent of a central simple algebra over the field of rational functions over a finite field. The algebra is given by a set of structure constants. The problem is reduced to the computation of a division algebra Brauer equivalent to the central simple algebra. This division algebra is constructed as a cyclic algebra, once the Hasse invariants have been computed. We give an application to skew constacyclic convolutional codes.

Explicit isomorphisms of quaternion algebras over quadratic global fields

“…In [23] and [19] one of the main techniques is to use an explicit bound on the number of monic irreducible polynomials in a given residue class [41]:…”

“…Namely, one chooses a uniformly random polynomial from that residue class (of a prescribed degree) and iterates until finding an irreducible polynomial (irreducibility can be checked with Berlekamp's algorithm [1]). A detailed analysis of this method can be found in both [23] and [19].…”

“…There are no known algorithms for subroutine 2 in the rational case. In the rest of this section we propose a polynomial -time algorithm for this task which is analogous to [19]. The key ingredient of the algorithm is a result by Schwinning [37] (which is referred to and generalized in [2]): Fact 5.5.…”

“…The task is to decide whether A and B are isomorphic, and if so, find an explicit isomorphisms between them. A special case of this problem when B = M n (K) is referred to as the explicit isomorphism problem which has various applications in arithmetic geometry [7], [14], [16], computational algebraic geometry [10] and coding theory [20], [19]. In 2012, Ivanyos, R ónyai and Schicho [26] proposed an algorithm for the explicit isomorphisms problem in the case where K is an algebraic number field.…”

Idempotents of $2\times 2$ matrix rings over rings of formal power series

“…The results of Foster and Steger were generalized also for matrices over noncommutative rings, see Song and Guo [14]. In [7] Gómez-Torrecillas, Kutas, Lobillo and Navarro presented an algorithm for of computing a primitive idempotent of a central simple algebra over the field F q (x) of rational functions over the finite field F q with applications to coding theory. In 1967 J.A.…”