New lower bounds on the minimum distance of quasi-twisted codes over finite fields are proposed. They are based on spectral analysis and eigenvalues of polynomial matrices. They generalize the Semenov-Trifonov and Zeh-Ling bounds in a manner similar to how the Roos and shift bounds extend the BCH and HT bounds for cyclic codes.
Spectral bounds on the minimum distance of quasi-twisted codes over finite fields are proposed, based on eigenvalues of polynomial matrices and the corresponding eigenspaces. They generalize the Semenov-Trifonov and Zeh-Ling bounds in a way similar to how the Roos and shift bounds extend the BCH and HT bounds for cyclic codes. The eigencodes of a quasi-twisted code in the spectral theory and the outer codes in its concatenated structure are related. A comparison based on this relation verifies that the Jensen bound always outperforms the spectral bound under special conditions, which yields a similar relation between the Lally and the spectral bounds. The performances of the Lally, Jensen and spectral bounds are presented in comparison with each other.
Polyadic constacyclic codes over finite fields have been of interest due to their nice algebraic structures, good parameters, and wide applications. Recently, the study of Type-I [Formula: see text]-adic constacyclic codes over finite fields has been established. In this paper, a family of Type-II [Formula: see text]-adic constacyclic codes is investigated. The existence of such codes is determined using the length of orbits in a suitable group action. A necessary condition and a sufficient condition for a positive integer [Formula: see text] to be a multiplier of a Type-II [Formula: see text]-adic constacyclic code are determined. Subsequently, for a given positive integer [Formula: see text], a necessary condition and a sufficient condition for the existence of Type-II [Formula: see text]-adic constacyclic codes are derived. In many cases, these conditions become both necessary and sufficient. For the other cases, determining necessary and sufficient conditions is equivalent to the discrete logarithm problem which is considered to be computationally intractable. Some special cases are investigated together with examples of Type-II polyadic constacyclic codes with good parameters.
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