In this paper, we study twisted group codes over finite groups by introducing a novel symmetric and non-degenerate bilinear form to define the α -dual of a code. Because the Euclidean inner product is not generally G-invariant for twisted group algebras unless α = α ^-1 , our proposed twisted bilinear form provides a natural framework for studying these codes. Moreover, we prove that this duality preserves important properties such as the maximum distance separable (MDS) property. We further investigate α -linear complementary dual ( α -LCD) codes, establishing that they are generated by idempotents and providing necessary conditions for such codes to be MDS. Additionally, we study the behavior of twisted duals under cohomologous 2-cocycles, showing that the resulting structures are equivalent. Finally, we present a decoding algorithm for α -LCD codes based on this new framework.
We investigate cyclic 𝔽_q -linear codes over the alphabet 𝔽_q𝔽_q^2 , where q is a prime power. First, its generator polynomials and minimal spanning set are determined. Then, examples of cyclic 𝔽_q -linear 𝔽_q^2 -codes that satisfy the well-known Singleton bound are constructed. Using a Gray map, we produce certain optimal linear codes over 𝔽_3 . Finally, we obtain a few optimal ternary linear complementary dual (LCD) codes from certain 𝔽_3 -linear 𝔽_3𝔽_9 -codes.
