Two Classes Of Optimal Restricted Strong Partially Balanced2-designs

Pei, Li, Wang and Safavi-Naini formulated restricted strong partially balanced t-designsfor the investigation of authentication codes with arbitration. Let v,b, μ,k, λ,t be pos-itive integers and t≤μ,(X, B) is a restricted partially balanced t-design, where X is av set, B is a collection of subsets of X with sizeμ×k such that the following propertiesare satisfied: every block B∈B is expressed as a disjiont union ofμsubblocks of sizek: B=B1∪B2∪···∪Bμ,Bi∩Bj=(i j, i, j∈{1,2,..., μ}); every t subset{x1,x2,...,xt} of X either occurs together in exactlyλblocks, B=B1∪B2∪···∪Bμ,x1∈Bi1,x2∈Bi2...,xt∈Bit, or dose not occur in any block, which is denoted as RPBD t (v,b, μ×k;λ,0).If a restricted partially balanced t-design RPBD t (v,b, μ×k;λ,0)is a restricted partially balanced s-design for0<s<t as well, then it is called a restrict-ed strong partially balanced t-design and is denoted as RSPBD t (v,b, μ×k;λ,0).Du and Liang proved that optimal splitting authentication codes that are multi-fold perfec-t against spoofing can be characterized in terms of restricted strong partially balancedt-designs, and have proved the existence of optimal restricted strong partially balanced2-designs ORSPBD2(v,b, μ×k;1,0) forμ×k=2×2,2×3,2×4and μ×k=3×2, v≡0(mod2).This paper mainly researches the existence of optimal strong partially balanced2-designs ORSPBD2(v,b,3×2;1,0) for v≡1(mod2) and the existence of optimalstrong partially balanced2-designs ORSPBD2(v,b,4×2;1,0) for v≡0(mod4).