Codes over an infinite family of algebras
نویسندگان
چکیده
In this paper, we will show some properties of codes over the ring Bk = Fp[v1, . . . , vk]/(v i = vi, ∀i = 1, . . . , k). These rings, form a family of commutative algebras over finite field Fp. We first discuss about the form of maximal ideals and characterization of automorphisms for the ring Bk. Then, we define certain Gray map which can be used to give a connection between codes over Bk and codes over Fp. Using the previous connection, we give a characterization for equivalence of codes over Bk and Euclidean self-dual codes. Furthermore, we give generators for invariant ring of Euclidean self-dual codes over Bk through MacWilliams relation of Hamming weight enumerator for such codes.
منابع مشابه
Codes as Ideals Over Some Pointed Hopf Algebras
We give a Decomposition Theorem for a family of Hopf algebras containing the well-know family of Taft Hopf algebras. Therefore, those indecomposable codes over this family of algebras (cf. [4]) is an indecomposable code over the studied case. We use properties of Hopf algebras to show that dual (in the module sense) of an ideal code is again an ideal code.
متن کاملThe Rational Schur Algebra
We extend the family of classical Schur algebras in type A, which determine the polynomial representation theory of general linear groups over an infinite field, to a larger family, the rational Schur algebras, which determine the rational representation theory of general linear groups over an infinite field. This makes it possible to study the rational representation theory of such general lin...
متن کاملar X iv : 0 81 1 . 04 21 v 1 [ qu an t - ph ] 4 N ov 2 00 8 QUANTUM ERROR CORRECTION ON INFINITE - DIMENSIONAL HILBERT SPACES
We present a generalization of quantum error correction to infinite-dimensional Hilbert spaces. The generalization yields new classes of quantum error correcting codes that have no finite-dimensional counterparts. The error correction theory we develop begins with a shift of focus from states to algebras of observables. Standard subspace codes and subsystem codes are seen as the special case of...
متن کاملNonexpansive mappings on complex C*-algebras and their fixed points
A normed space $mathfrak{X}$ is said to have the fixed point property, if for each nonexpansive mapping $T : E longrightarrow E $ on a nonempty bounded closed convex subset $ E $ of $ mathfrak{X} $ has a fixed point. In this paper, we first show that if $ X $ is a locally compact Hausdorff space then the following are equivalent: (i) $X$ is infinite set, (ii) $C_0(X)$ is infinite dimensional, (...
متن کاملTensor products of ideal codes over Hopf algebras
We study indecomposable codes over the well-known family of Radford Hopf algebras. We use properties of Hopf algebras to show that tensors of ideal codes are ideal codes, extending the corresponding result given in [4] and showing that in this case, semisimplicity is lost.
متن کامل