🤖 AI Summary
This study addresses the systematic classification of centrally symmetric alternant codes induced by generalized Reed–Solomon codes. By analyzing their inherited automorphism structure, the authors introduce a twisted conjugation action associated with projective semilinear transformations and, for the first time, employ Shintani’s theorem to establish a correspondence between this twisted action and ordinary conjugacy in the projective semilinear group. Leveraging an analysis of γ_{p^j}-similarity classes, the paper derives necessary and sufficient conditions for an alternant code to admit a centrally symmetric structure induced by a projective semilinear automorphism. This result yields a unified, automorphism-based classification framework that enables the systematic categorization of all such induced centrally symmetric alternant codes.
📝 Abstract
This paper presents a classification of induced centrosymmetric alternant codes through the study of the automorphism structures inherited from Generalized Reed--Solomon (GRS) codes. We introduce the twisted conjugation action naturally associated with projective semilinear transformations and establish its correspondence with ordinary conjugacy in the projective semilinear group. This correspondence enables the application of Shintani's theorem to classify the $γ_{p^j}$-similarity classes of involutions. As a consequence, we obtain necessary and sufficient conditions for an alternant code to admit a centrosymmetric structure induced by a projective semilinear automorphism. The resulting classification unifies the different families of induced centrosymmetric alternant codes within a common automorphism-based framework.