The Upper Bounds And Constructions Of Separable Codes And Strongly Separable Codes

With the advancement of communication technology,multimedia products bring great economic benefits to people.However,the pirate copies are increasingly serious problems in copyright protection of multimedia contents.In order to against pirate copies,Cheng and Miao proposed a concept of a logical anti-collusion code(LACC)in 2011.Then Cheng,Jiang et al.proved that separable codes and strongly separable codes can be used to construct LACCs with different tracing complexities.So it is meaningful to study separable codes and strongly separable codes.First the following notations are very useful.Let n,M and q be positive integers,Q = {0,1,…,q-1}.Let C be an(n,M,q)code,For any subset of codewords CO(?)C C,we define the descendant of C0 by desc(C0)= {(x(1),x(2),…,x(n))T ?Qn|x(i)?C((i),1?i?n},where C0(i)={c(i)?Q |c =(c(1),c(2),…,c(n))T? C0}.Definition 1 Let C be an(n,M,q)code and t?2 be an integer.· C is a t-separable code,or t-SC(n,M,q),if for any C1,C2(?)C such that 1?C1|?t,1?|C2|?t and C1?C2,we have desc(C1)? desc(C2),that is there is at least one coordinate i?i?n,such that C1(i)?C2(i).· C is a strongly t-separable code,or t-SSC(n,M,q),if for any C0(?)C such that 1?|C0|?t,we have ?C'?S(C0)C' =C0,where S(C0)= {C'(?)C|desc(C')= desc(C0)}.· C is a t-frameproof code,or t-FPC(n,M,q),if for any C'(?)C such that |C'|?t,it holds that desc(C')?C=C'.that is,for any c =(c(1),…,c(n))T?C\C',there is at least one coordinate i,1?i?n,such that c(i)(?)C'(i).Definition 2 C is a(a,b;c)-(n,M,q)code,or(a,b;c)code,if for any C1,C2(?)C,|C1|= a,|C2|=b,|C1?C2|=c,we have desc(C1)?desc(C2).That is,there is at least one coordinate i,1?i?n,such that C1(i)?C2(i).In this thesis,we obtained the following main results.Theorem 1 C is(2,2;0)-(n,M + 2,q)if and only if C is 2-SC(n,M,q).Theorem 2 There exist a 2-SC(n,M,q)with M?(?-2n-3?4)q2n/3-2,where 0???21-n/3.Theorem 3 There exits a 2-SC(n,M,q)withTheorem 4 Let q?2,n?2,q,n,N be integers,there exits a 2-SC(n,M,q),whereTheorem 5 A(4,M,q)code is a 4-SC if only if the following two conditions hold.1)C is a 3-FPC(4,M,g):2)Configurations ?1,?2,?3,?4,?5,?6,?7 and their conjugates are the forbidden configurations of C.· In ?1,a1?c1?d1,a2?b2?c2,a3?b3?d3,a4?b4,a4?c4,a4?d4.· In ?2,a1?b1?d1,a2?b2?c2,a3?b3?d3,a4?b4?c4.· In ?3,a1?b1?d1,a2?c2,a3?b3?d3,a4?b4?c4,a4?d4,b4?d4.· In ?4??7,|{ai,bi,ci,di}|=4,where 1?i?4.Theorem 6 For any q?3,a(3,M,q)code is a 3-SSC(3,M,q)if and only if it is a 3-SC(3,M,q).Theorem 7 For any prime power q=q16,where q1= 6t+ 1,there exists a 3-SSC(3,M,q),where M = ?(q5/3 + q4/3-q).This thesis is organized as follows:Related definitions and main results are presented in Chapter 1.Several new lower bounds on 2-SC(t,M,q)will be derived in Chapter 2.In Chapter 3,forbidden configurations of 4-SC(4,M,q)will be proposed.In Chapter 4,we will derive a new lower bounds on 3-SSC(3,M,q).Conclusions and research problems are given in Chapter 5.