Quasi-Cyclic Perfect Codes in Doob Graphs and Special Partitions of Galois Rings
Minjia Shi, Xiaoxiao Li, Denis S. Krotov, Ferruh Özbudak · IEEE Transactions on Information Theory · 2023
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}$.