# Schur ring and Codes for $S$-subgroups over $\Z_{2}^{n}$

## TL;DR

This paper explores the connection between $S$-subgroups in $ extbf{Z}_2^n$ and binary codes, revealing conditions under which these subgroups are free and their invariance properties under permutation actions.

## Contribution

It establishes the relationship between $S$-subgroups and specific classes of binary codes, detailing conditions for freeness and invariance under permutation groups.

## Key findings

- $S$-subgroups are free when codes are both $P(T)$-codes and $G$-codes.
- Constructed codes are cyclic, decimated, or symmetric, leading to specific invariance properties.
- No codes generate the entire $ extbf{Z}_2^n$ in any $ extbf{G}_n(a)$-complete $S$-set.

## Abstract

In this paper the relationship between $S$-subgroups in $\Z_{2}^{n}$ and binary codes is shown. If the codes used are both $P(T)$-codes and $G$-codes, then the $S$-subgroup is free. The codes constructed are cyclic, decimated or symmetric and the $S$-subgroups obtained are free under the action the cyclic permutation subgroup, invariants under the action the decimated permutation subgroup and symmetric under the action of symmetric permutation subgroup, respectively. Also it is shows that there is no codes generating whole $\Z_{2}^{n}$ in any $\G_{n}(a)$-complete $S$-set of the $S$-ring $\mathfrak{S}(\Z_{2}^{n},S_{n})$.