A method for constructing splitting (v,c u, ) BIBDs. Stela Zhelezova Institute of Mathematics and Informatics, BAS

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1 A method for constructing splitting (v,c u, ) BIBDs Stela Zhelezova Institute of Mathematics and Informatics, BAS

2 Motivation T O m = e(s) a model of G.J.Simmons, 1982 R 3-splitting (2,7,7) A-code (S,M,E) A-code S source state space E key space М message space, m = e(s) splitting strategy to determine m M e(s) =c for е Е, s S е Е, s s 1, е(s) e(s 1 )=0 S 1 S 2 е 1 {2,3,4} {5,6,7} е 2 {1,2,6} {3,4,7} е 3 {1,3,5} {2,4,6} е 4 {1,4,7} {2,3,5} е 5 {3,6,7} {1,4,5} е 6 {2,5,7} {1,3,6} е 7 {4,5,6} {1,2,7}

3 Introduction 2-(v,k,λ) design: V = {P i } v i=1 B = {B j } b j=1 finite set of v points; finite collection of b blocks: k-element subsets of V; D = (V, B ) 2-(v,k,λ) design if any 2-element subset of V is in λ blocks of B; r(k-1) = λ(v-1) bk = vr

4 Introduction Isomorphic designs exists a one-to-one correspondence between the point sets and block families of both designs, which does not change the incidence. Automorphism permutation of the points that transforms the blocks into blocks.

5 Introduction Parallel class partition of the design point set by blocks. Resolution partition of the family of blocks by parallel classes. Isomorphic resolutions - exists an automorphism of the design transforming each parallel class of the first resolution into a parallel class of the second one.

6 Introduction Ogata, Kurosawa, Stinson, Saido, 2004; (v, c u, )-splitting BIBD (V,B) with the next properties: V = v finite set of points p i, 1 i v ; B = b finite family of super-blocks, B j V, B j B, 1 j b; B j B, B j = B ij,1 B j,u, B j,1 = = B j,u =c; p x, p y V, x y, exactly super-blocks B j, x B i,m, y B i,n, m n. splitting BIBDs splitting authentication codes

7 Introduction (v, c u, )-splitting BIBD (V,B) with the next properties: V = v finite set of points p i, 1 i v ; B = b finite family of super-blocks, B j V, B j B, 1 j b; B j B, B j = B ij,1 B j,u, B j,1 = = B j,u =c; p x, p y V, x y, exactly super-blocks B j, x B i,m, y B i,n, m n. r = b = (v 1) c(u 1) v(v 1) c 2 u(u 1) v u. c v 1 0 mod (c(u 1)) v v 1 0 mod (c 2 u(u 1))

8 Introduction Equivalent splitting BIBDs exists a permutation of the points which transforms each super-block of the first splitting design to a super-block of the second one.

9 Existence results: Du, B.: Splitting balanced incomplete block designs with block size 3 2, 2004 Ge G., Miao Y., Wang L., Combinatorial constructions for authentication codes, 2005 Su, R., Wang, J.: On the existence of (v, 3 3, )-splitting balanced incomplete block designs with between 2 to 9, 2008 Su, R., Wang, J.: Further Results on the Existence of Splitting BIBDs and Application to Authentication Codes, 2010

10 Incidence matrix of a resolvable BIBD (6,3,4) BIBD, b=20, r= A=(a ij ) v b a ij = 1, if P i B j a ij = 0, if P i B j i = 1,2,,v, j=1,2,,b Parallel class partition of the point set by blocks. Resolution partition of the collection of design s blocks by parallel classes.

11 Incidence matrix of a resolvable BIBD (6,3,4) BIBD, b=20, R=10 (10,6,6) (qk,k, )BIBD (r,qk,r- ) q equidistant code, q>1 Semakov, Zinoviev, 1968

12 Construction method (7, 3 2, 3)-splitting BIBD incidence super-matrix A=(a ij ) v b a ij = 1, if p i B j,k a ij = 0, if p i B j,k i = 1,2,,v, j=1,2,,b k=1,, u

13 Construction method Construction of the v b incidence super-matrix point by point: alphabet with u elements {1,2,...,u} each symbol is exactly c times in each column and r = (v 1)/(c(u 1)) in each row; each pair of points has exactly times different symbols in each coordinate; exactly v-c.u holes (0) in each coordinate; Minimality test after each point. (7, 3 2, 3)- splitting BIBD

14 Construction method (7, 3 2, 3)- splitting BIBD splitting (2,7,7) A-code S 1 S 2 е 1 {2,3,4} {5,6,7} е 2 {1,2,6} {3,4,7} е 3 {1,3,5} {2,4,6} е 4 {1,4,7} {2,3,5} е 5 {3,6,7} {1,4,5} е 6 {2,5,7} {1,3,6} е 7 {4,5,6} {1,2,7}

15 Results v c u r b Number

On the Classification of Splitting (v, u c, ) BIBDs

On the Classification of Splitting (v, u c, ) BIBDs BULGARIAN ACADEMY OF SCIENCES CYBERNETICS AND INFORMATION TECHNOLOGIES Volume 18, No 5 Special Thematic Issue on Optimal Codes and Related Topics Sofia 2018 Print ISSN: 1311-9702; Online ISSN: 1314-4081