Computing separability elements for the sentence-ambient algebra of split ideal codes

Cyclic structures on convolutional codes are modeled using an Ore extension A [ z ; σ ] of a finite semisimple algebra A over a finite field F . In this context, the separability of the ring extension F [ z ] ⊂ A [ z ; σ ] implies that every ideal code is a split ideal code. We characterize this separability by means of σ being a separable automorphism of the F -algebra A. We design an algorithm that decides if such a given automorphism σ is separable. In addition, it also computes a separability element of F [ z ] ⊂ A [ z ; σ ] , which is important because it can be used to find an idempotent generator of each ideal code with sentence-ambient A [ z ; σ ] .