Optimal Packings Avoiding Units Within Certain Distance And Codes For Crosstalk Avoidance

This dissertation consists of two parts.In the first part,we study the packing analog of balanced sampling plans avoiding adjacent units(BSA)which was first introduced in 1988 by Hedayat,Rao,and Stufken.BSAs are widely used in statistical designs of experiments.When the necessary conditions of the existence of BSAs are not satisfied,we consider packing sampling plans avoiding adjacent units(PSA).That is,every pair of distinct points in set X of order n with distance more than ? appears in at most one block of such PSA(n,k;?)s with blocks size k.In 2015,Chee et al.studied both(circular)CPSA(n,3,?)s and(linear)LPSA(n,3,?)s.For ?=1,they completely determined the size of optimal CPS As and optimal LPS As for all n.They also constructed optimal packings for n?2,3,4,5(mod 6)and ?=2.Zhang and Chang considered only balanced cases for optimal CPSA(n,3;3)s.We will solve all remaining cases of these optimal CPS As and optimal LPSAs for ?=2,3.Moreover,we give the characterization of leave graphs of such CPSAs and LPSAs.In the second part,we will study the maximum size of codes that simultaneously pro-vide for low power dissipation,crosstalk avoidance,and error correction in the ultra-deep submicron/nanometer VLSI fabrication,which were recently introduced in 2015 by Chee et al.They illustrated the worst crosstalk couplings into four types and revealed that such codes are closely related to balanced sampling plans avoiding adjacent units.They also constructed optimal or asymptotically optimal constant weight codes that eliminate each kind of crosstalk.They determined the size of optimal(n,4,3)-{?,?} code and asymptotically optimal(n,4,3)-? code for all n(?)0,1(mod 6).We will give the complete solution of these optimal and asymptotically optimal codes,by utilizing optimal packing sampling plans avoiding units within distance two.We also construct improved asymptotically optimal(n,4,3)-? code for all even orders n?14.Furthermore,for each odd order n? 3,we will show that an optimal weight three code avoiding the crosstalk type-? is also an optimal code avoiding the crosstalk of type-{?,?}.