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].