No abstract

Differential Convolutional Codes with designed Hamming distance are defined, and an algebraic decoding algorithm, inspired by Peterson-Gorenstein-Zierler's algorithm, is designed for them.

Roos’ direct sum decomposition of a cyclic convolutional code based upon a suitable construction of a minimal encoder is extended to a broader class of convolutional codes endowed with additional algebraic structure. The two (equivalent) usual concepts of a convolutional code are taken into account, as a vector space and as a module.

A large class of MDS linear codes is constructed. These codes are endowed with an efficient decoding algorithm. Both the definition of the codes and the design of their decoding algorithm only require from Linear Algebra methods, making them fully accesible for everyone. Thus, the first part of the paper develops a direct presentation of the codes by means of parity-check matrices, and the decoding algorithm rests upon matrix and linear maps manipulations. The somewhat more sophisticated mathematical context (non-commutative rings) needed for the proof of the correctness of the decoding algorithm is postponed to the second part. A final section locates the Reed-Solomon skewdifferential codes introduced here within the general context of codes defined by means of skew polynomial rings.

We give necessary and sufficient conditions on an Ore extension A[x; σ, δ], where A is a finite dimensional algebra over a field F, for being a Frobenius extension over the ring of commutative polynomials F [x]. As a consequence, as the title of this paper highlights, we provide a negative answer to a problem stated by Caenepeel and Kadison.