Primitive idempotents in central simple algebras over \(\mathbb{F}_q(t)\) with an application to coding theory (Q2667078)

Let \(\mathcal{A}\) be a central simple algebra over \(\mathbb{F}_q(t)\) of dimension \(n^2\) given by structure constants. By Wedderburn's theorem, it is isomorphic to a matrix algebra \(M_k(D)\) for a unique division algebra \(D\). The authors describe, under some assumptions, a randomised polynomial-time algorithm that computes an explicit isomorphism \(\mathcal{A}\cong M_k(D)\). It is implicitly taken for granted in the article that this challenge is equivalent to finding a primitive (i.e.,\ not the sum of several orthogonal idempotents) idempotent \(e\in\mathcal{A}\), as one can then indeed take the left-action of \(\mathcal{A}\) on \(\mathcal{A} e\) and \(D=e\mathcal{A} e\). The algorithm steps are as follows (with assumptions made by the algorithm given in each step): \begin{itemize} \item[1.] Compute the Hasse invariants of \(\mathcal{A}\). To decide whether \(\mathcal{A}\) has Hasse invariant \(k/n\) at a place \(f\) dividing the discriminant, one constructs a division algebra \(D'\) with Hasse invariant \(\frac{n-k}{n}\) at \(f\) (this assumes coprime \(q\) and \(n\)) and checks whether the local index of \(\mathcal{A}\otimes D'\) is \(1\). \item[2.] Construct a division algebra \(D\) with the same Hasse invariants. This either uses the same previously known algorithm as for \(D'\) in Step 1 based on cyclic algebras, assuming that \(q\) and \(n\) are coprime and that \(\mathcal{A}\) is split at the infinite place (or split at some linear place by transformation of variables). Alternatively, the authors provide a new algorithm when \(\mathcal{A}\) is non-split at the infinite place based on symbol algebras (e.g.,\ quaternion algebras in rank \(2\)), assuming that \(q\equiv1\pmod n\) and so \(\mathbb{F}_q\) contains primitive \(n\)-th roots of unity. \item[3.] Construct an isomorphism \(\theta:\mathcal{A}\otimes M_k(D^{\mathrm{op}})\cong M_{n^2}(\mathbb{F}_q(t))\). This uses an algorithm from [\textit{G. Ivanyos} et al., Found. Comput. Math. 18, No. 2, 381--397 (2018; Zbl 1391.16032)] for the same question when \(D=\mathbb{F}_q(t)\), i.e.,\ the full matrix algebra case. \item[4.] Use \(\theta\) to compute an isomorphism \(\mathcal{A}\cong M_k(D)\). \end{itemize} The last section of the article gives an application to coding theory. Let \(\sigma\) be an automorphism of \(\mathbb{F}_q(t)\) order \(n\) and denote by \(\mathbb{F}_q(t)[x,\sigma]\) the (non-commutative) ring of skew-polynomial with respect to \(\sigma\). For non-zero \(\lambda\in \mathbb{F}_q(t)^\sigma\), a \textit{skew-constacyclic code} is a left ideal of \(\mathcal{A}=\mathbb{F}_q(t)[x,\sigma]/(x^n-\lambda)\), equipped with the Hamming weight with respect to the coefficients. If \(\lambda=1\) (or a norm of an element in \(\mathbb{F}_q(t)\)), this is the special case of skew-cyclic codes from [\textit{J. Gómez-Torrecillas} et al., IEEE Trans. Inf. Theory 62, No. 5, 2702--2706 (2016; Zbl 1359.94752)]. If \(\lambda\) is not a norm, let \(m\) be the index of \(\mathcal{A}\), \(e\in\mathcal{A}\) a primitive idempotent and fix \(k\leq n/m\). The authors show that the ideal generated by \(e,\sigma^m e,\dots,\sigma^{m(k-2)}e\) is a code with Hamming distance at least \(k\).