tailieunhanh - Examples of self-dual codes over some sub-Hopf algebras of the Steenrod algebra

Codes over the finite sub-Hopf algebras A(n) of the (mod 2) Steenrod algebra A were studied by Dougherty and Vergili. In this paper we study some Euclidean and Hermitian self-dual codes over A(n) by considering Milnor basis elements. | Turk J Math (2017) 41: 1313 – 1322 ¨ ITAK ˙ c TUB ⃝ Turkish Journal of Mathematics doi: Research Article Examples of self-dual codes over some sub-Hopf algebras of the Steenrod algebra ˙ I˙ ∗, Ismet ˙ Tane VERGIL KARACA ˙ Department of Mathematics, Ege University, Izmir, Turkey Received: • Accepted/Published Online: • Final Version: Abstract: Codes over the finite sub-Hopf algebras A(n) of the (mod 2) Steenrod algebra A were studied by Dougherty and Vergili. In this paper we study some Euclidean and Hermitian self-dual codes over A(n) by considering Milnor basis elements. Key words: Hopf algebra, Steenrod algebra, self-dual codes 1. Introduction The elements of the (mod 2) Steenrod algebra A are natural transformations between cohomology groups of topological spaces and useful tools for computing the homotopy groups of n -spheres. The finite subalgebras are determined by the profile functions h given in (4); each profile function constructs only one subalgebra [5]. Considering the function h(t) given in equation (5), we construct the subalgebras A(n) of A for all n ≥ 0. These subalgebras are nested, . A(n) is contained in A(m) if n < m , and their union is the entire algebra. Further, A(n) is a noncommutative Frobenius ring, and so the MacWilliams theorems hold and one can study codes over A(n) [8]. Dougherty and Vergili [2] studied the codes in A(n) by considering the Z -base system over A(n), which can be extended to the whole algebra A [10]. In this paper, we study codes over A(n) by changing the base system. We use the Milnor basis, which is constructed in A, compatible with A(n) , and provides a product formula for two Milnor basis elements. We examine a Euclidean and Hermitian self-dual code over A(1) and show the generalization of that code to A(n) is also a Euclidean and Hermitian self-dual. 2. Definitions and notations . The Steenrod algebra A The (mod