Doubly Resolvable Canonical Kirkman Packing Designs and its Applications

Abstract

Let $$v \equiv 4 \pmod 6$$ and $$v\ge 4$$ . A canonical Kirkman packing design of order v, denoted by $$\hbox {CKPD}(v)$$ , is a resolvable packing with $$r=(v-4)/2$$ parallel classes such that (i) each parallel class consists of a size 4 block and $$(v- 4)/3$$ triples; (ii) the leave consists of the union of $$(v- 4)/2$$ vertex-disjoint edges and a $$K_4$$ with no vertices in common with those edges. A canonical Kirkman packing design is said to be doubly resolvable if there exist a pair of orthogonal resolutions. A doubly resolvable packing design is the generalization of the Kirkman square, which is inextricably bound up with the existence of some constant weight codes such as constant composition codes and permutation codes, etc. In this paper, we establish the spectra of doubly resolvable $$\hbox {CKPD}(v)\hbox {s}$$ with 36 possible exceptions for v. As its direct application, a class of permutation codes are obtained.