Applications Of Langford Sequence In Constructing Two Kinds Of Combinatorial Structure

Mentioned the statistical problems in environmental and ecological resources research, the research populations is sequentially ordered usually and neighboring units may provide similar information. When sampling the quantitative characteristics of the populations, the selection of dispersed units may result in increasing efficiency. Balanced sampling plans excluding contiguous units (or BSEC) were first introduced by Hedayat, Rao and Stufken in1988based on this background.In this note, we use special Langford sequences to construct balanced sampling plans excluding contiguous units with kites (cyclic kite-BSEC (v,λ)) and cyclic balanced sampling designs avoiding adjacent units with block size three and α≤4(cyclic BSA(v,3,λ;4)). Then obtain the necessary and sufficient conditions for their existence.The necessary and sufficient conditions for the existence of cyclic kite-BSEC is: v≥11and λ(v-3)≡0(mod8).The necessary and sufficient conditions for the existence of CBSA(v,3,λ;4) is: v≥27and (1) v=3(mod6) when λ≡1,5(mod12);(2) v≡0(mod3) and v≠2(mod4) when λ≡2,10(modl2);(3) v=1(mod2) when λ=3(mod6);(4) v=0(mod3) when A=4,8(mod12);(5) v≠2(mod4) when λ=6(mod12);(6) v is unrestricted when λ=0(mod12).