Dual skew codes from annihilators: Transpose Hamming ring extensions

Linear codes may be endowed with cyclic structures by means of skew polynomial rings. This is the case of Piret cyclic convolutional codes [26] and the subsequent generalizations and alternatives (see [27], [14], [11], [25], [16], [20]). Non commutative cyclic structures of this kind have been also considered for block linear codes ([7], [5], [4], [12], [1]), and for linear codes over commutative rings ([6], [22], [10]). A desirable property of any class of linear codes is to be stable under duals. This property has been already studied in several of the aforementioned references. A common feature of many of these approaches to duality is the presence of a suitable anti-isomorphism of rings that encodes, more or less explicitly, the transfer of the cyclic structure from the code to its dual. Our aim is to present a systematization of this method, besides some relevant examples where it successfully applies. The strategy is to establish a formal framework, called transpose Hamming ring extension, designed to derive that the dual of every cyclic code is cyclic. Cyclic codes will be, from an algebraic point of view, identified as left ideals of suitable (noncommutative) ring extensions of a given commutative ring C, well understood that such an “identification” has to be made explicit by an isomorphism of C–modules from the ring to C, where m is the length of the C–linear code. The transposition will be an anti-isomorphism of rings which allows to transform annihilators into duals. Details are to be found in Section 1 In Section 2 we apply our general approach to left ideal convolutional codes in the sense of [25], extending to a more general setting, and improving, results from [25] and [16] on the description of dual codes in this setting (Theorem 11). The case of a simple word-ambient algebra is analyzed in detail (Theorem 23). Section 3 is devoted to dual codes of skew constacyclic codes over a commutative ring. Several results from [7], [6], [22], [10] on these codes are covered by our general result (Theorem 29). Finally, in Section 4, we compute (Theorem 37) the dual of a skew Reed Solomon code over a general field in the sense of [18], and, as a consequence, these codes are shown to be evaluation codes (Theorem 39).