Quasi-cyclic perfect codes in Doob graphs and special partitions of Galois rings

Abstract

The Galois ring GR$(4^{\Delta })$ is the residue ring $Z_4[x]/(h(x))$, where $h(x)$ is a basic primitive polynomial of degree $\Delta$ over $Z_4$. For any odd $\Delta$ larger than $1$, we construct a partition of GR$(4^{\Delta })\backslash \{0\}$ into $6$-subsets of type $\{a,b,-a-b,-a,-b,a+b\}$ and $3$-subsets of type $\{c,-c,2c\}$ such that the partition is invariant under the multiplication by a nonzero element of the Teichmuller set in GR$(4^{\Delta })$ and, if $\Delta$ is not a multiple of $3$, under the action of the automorphism group of GR$(4^{\Delta })$. As a corollary, this implies the existence of quasi-cyclic additive $1$-perfect codes of index $(2^{\Delta }-1)$ in $D((2^{\Delta }-1)(2^{\Delta }-2)/6,2^{\Delta }-1)$ where $D(m,n)$ is the Doob metric scheme on $Z^{2m+n}$.

Accepted version; 7 IEEE TIT pages

Cited by in corpus (2)

• Quasi-cyclic codes of index 2
• Triangle decompositions of PG(n-1,2)