On the existence of self-dual permutation codes of finite groups

Abstract

Motivated by a research on self-dual extended group codes, we consider permutation codes obtained from submodules of a permutation module of a finite group of odd order over a finite field, and demonstrate that the condition "the extension degree of the finite field extended by n'th roots of unity is odd" is sufficient but not necessary for the existence of self-dual extended transitive permutation codes of length n + 1. It exhibits that the permutation code is a proper generalization of the group code, and has more delicate structure than the group code.

Metabolic modules of finite group algebras over finite fields of characteristic two

“…To get a faithful F G-module, we consider the regular module F G. By [7, Exercise 24, page 120], we may choose a G-invariant non-degenerate symplectic form on F G so that it becomes into a faithful symplectic F G-module. It is easy to see that any nontrivial element in G of odd order has the characteristic polynomial on F G of the form (x p − 1) 4 , and hence Theorem A implies that the regular module F G is hyperbolic. Now let V = F G⊥U be the orthogonal direct sum of these two symplectic F G-modules.…”

“…The hyperbolic symplectic modules are quite useful for investigating correspondences of characters, extensions of characters, monomial characters and Mgroups, for example, see [2,10,14,15,16,19,24,25]. The study of hyperbolic symmetric modules is used to determine the existence of self-dual codes and to characterize the automorphisms of self-dual codes, see [3,4,6,17,18]. Also, in order to study the Frobenius-Schur indicators of group representations, J.G.…”

Iso-orthogonality and Type-II duadic constacyclic codes

“…By appending with one bit, self-dual extended cyclic codes might be obtained. It is the same for group codes, more generally, for transitive permutation codes, see [13].…”

“…4,10,13,16,19, 34, 40, 52}; then sP = Q 4 ∪ Q 5 ∪ Q 6 = {22, 25, 31, 37, 43, 46, 55, 58, 61}. ∪ P ∪ sP = P n,λ , P (0) n,λ ∩ P = P (0) n,λ ∩ sP = P ∩ sP = ∅.…”

Galois self-dual constacyclic codes

“…Brualdi and Pless [3], Ward and Zhu [24], Ling and Xing [17], Sharma, Bakshi, Dumir and Raka [22] studied the polyadic cyclic or abelian codes. Williams [23], Matinnes-Pérez and Williams [18], Fan and Zhang [10], Jitman, Ling and Solé [14] studied self-dual or Hermitian self-dual group codes.…”