Generating Idempotents in Ideal Codes
Most of the codes used in engineering support a vector space structure (linear block codes) or becomes a submodule of a polynomial ring (convolutional codes). In the linear case, the benefits are amplified if we also consider the notion of cyclicity, since the ambient space is also endowed with an algebra structure and cyclic codes come to be ideals. Over convolutional codes, this notion requires something more sophisticated than a simple extension of the block one [1], and the underlying working algebra is no longer a polynomial ring but an skew polynomial ring A[z;σ] over a finite commutative semisimple algebra A. Very recently, in [3], these codes are called ideal codes and they are defined over Ore polynomial rings A[z;σ, δ], where A is a finite (possibly non-commutative) semisimple algebra. Nevertheless, effective algorithms are provided whenever A is a separable group algebra of a finite group over a finite field. In this work we aim to cover more examples, see Example 3, than the former papers by assuming certain mild conditions of separability in A, see Corollary 2. In particular, we prove that they are generated by an idempotent element. We also provide an algorithm for computing a generating idempotent, which in particular is applicable the ideal codes from [1]. For brevity, we only shall consider Ore polynomial rings A[z;σ, δ] where the σ-derivation δ = 0, albeit, under suitable conditions, our results remain true for an arbitrary δ. In our examples, except for 0 and 1, we shall write the elements of a finite field F as powers of a primitive element.