Enumeration techniques on cyclic Schur rings

Enumeration of Schur Rings Over Small Groups

Enumeration and construction of additive cyclic codes over Galois rings

Let R = GR(p, l) be a Galois ring of characteristic p and cardinality pεl, where p and l are prime integers. First, we give a canonical form decomposition for additive cyclic codes over R. This decomposition is used to construct additive cyclic codes and count the number of such codes, respectively. Thenwe give the trace dual code for each additive cyclic code over R from its canonical form dec...

Automorphism Groups of Schur Rings

In 1993, Muzychuk [18] showed that the rational Schur rings over a cyclic group Zn are in one-to-one correspondence with sublattices of the divisor lattice of n, or equivalently, with sublattices of the lattice of subgroups of Zn. This can easily be extended to show that for any finite group G, sublattices of the lattice of characteristic subgroups of G give rise to rational Schur rings over G ...

Schur-finiteness in Λ-rings

We introduce the notion of a Schur-finite element in a λ-ring. Since the beginning of algebraic K-theory in [G57], the splitting principle has proven invaluable for working with λ-operations. Unfortunately, this principle does not seem to hold in some recent applications, such as the K-theory of motives. The main goal of this paper is to introduce the subring of Schur-finite elements of any λ-r...

The Number of Indecomposable Schur Rings over a Cyclic 2-group

Indecomposable Schur rings over a cyclic group Zn of order n are considered. In the case n = p, p an odd prime, the total number of such rings was described in terms of Catalan numbers by Liskovets and Pöschel [Discr. Math. 214 (2000), 173–191]. Here, a closed formula is shown for the total number of indecomposable Schur rings over Z2m using Catalan and Schröder numbers. The result is obtained ...