Orbit matrices of symmetric designs and related self-dual codes Orbit matrices of symmetric designs and (a joint work with Dean Crnkovi´ c) related self-dual codes Sanja Rukavina Introduction sanjar@math.uniri.hr Orbit matrices of symmetric Department of Mathematics designs University of Rijeka, Croatia Codes from orbit matrices of symmetric designs Symmetries and Covers of Discrete Objects Self-dual 14 - 19 February 2016, Queenstown, New Zealand codes from extended orbit matrices Supported by CSF under the project 1637. 1 / 29
Orbit matrices of symmetric designs and 1 Introduction related self-dual codes Orbit matrices of symmetric designs Introduction Orbit matrices of symmetric designs Codes from 2 Codes from orbit matrices of symmetric designs orbit matrices of symmetric designs Self-dual codes from extended orbit 3 Self-dual codes from extended orbit matrices matrices 2 / 29
Symmetric designs A t − ( v , k , λ ) design is a finite incidence structure Orbit matrices D = ( P , B , I ) satisfying the following requirements: of symmetric designs and 1 |P| = v , related self-dual codes 2 every element of B is incident with exactly k elements of Introduction P , Orbit matrices of symmetric designs 3 every t elements of P are incident with exactly λ elements Codes from of B . orbit matrices of symmetric Every element of P is incident with exactly r elements of B . designs The number of blocks is denoted by b . Self-dual codes from If |P| = |B| (or equivalently k = r ) then the design is called extended orbit matrices symmetric . The incidence matrix of a design is a b × v matrix [ m ij ] where b and v are the numbers of blocks and points respectively, such that m ij = 1 if the point P j and the block x i are incident, and m ij = 0 otherwise. 3 / 29
Tactical decomposition Orbit matrices of symmetric designs and Let A be the incidence matrix of a design D = ( P , B , I ). A related self-dual codes decomposition of A is any partition B 1 , . . . , B s of the rows of A (blocks of D ) and a partition P 1 , . . . , P t of the columns of A Introduction Orbit matrices (points of D ). of symmetric designs Codes from For i ≤ s , j ≤ t define orbit matrices of symmetric designs α ij = |{ P ∈ P j | P I x }| , for x ∈ B i arbitrarily chosen, Self-dual β ij = |{ x ∈ B i | P I x }| , for P ∈ P j arbitrarily chosen. codes from extended orbit matrices We say that a decomposition is tactical if the α ij and β ij are well defined (independent from the choice of x ∈ B i and P ∈ P j , respectively). 4 / 29
Automorphism group Orbit matrices of symmetric designs and An isomorphism from one design to other is a bijective related self-dual codes mapping of points to points and blocks to blocks which preserves incidence. An isomorphism from a design D onto Introduction Orbit matrices itself is called an automorphism of D . The set of all of symmetric designs automorphisms of D forms a group called the full Codes from orbit matrices automorphism group of D and is denoted by Aut ( D ). of symmetric designs Let D = ( P , B , I ) be a symmetric ( v , k , λ ) design and Self-dual codes from G ≤ Aut ( D ).The group action of G produces the same number extended orbit matrices of point and block orbits. We denote that number by t , the G − orbits of points by P 1 , . . . , P t , G − orbits of blocks by B 1 , . . . , B t , and put |P r | = ω r , |B i | = Ω i , 1 ≤ i , r ≤ t . 5 / 29
The group action of G induces a tactical decomposition of Orbit matrices of symmetric the incidence matrix of D . Denote by γ ij the number of points designs and related of P j incident with a representative of the block orbit B i . For self-dual codes these numbers the following equalities hold: Introduction Orbit matrices of symmetric t designs � γ ij = k , (1) Codes from orbit matrices j =1 of symmetric designs t Ω i � Self-dual γ ij γ is = λω s + δ js · n , (2) codes from ω j extended orbit i =1 matrices where n = k − λ is the order of the symmetric design D . 6 / 29
Orbit matrix Orbit matrices Definition 1 of symmetric designs and related A ( t × t )-matrix M = ( γ ij ) with entries satisfying conditions (1) self-dual codes and (2) is called an orbit matrix for the parameters ( v , k , λ ) Introduction and orbit lengths distributions ( ω 1 , . . . , ω t ), (Ω 1 , . . . , Ω t ). Orbit matrices of symmetric designs Codes from orbit matrices Orbit matrices are often used in construction of designs with a of symmetric designs presumed automorphism group. Construction of designs Self-dual admitting an action of the presumed automorphism group codes from extended orbit consists of two steps: matrices 1 Construction of orbit matrices for the given automorphism group, 2 Construction of block designs for the obtained orbit matrices. 7 / 29
Codes from orbit matrices of symmetric designs Orbit matrices of symmetric designs and related self-dual codes Theorem 1 [M. Harada, V. D. Tonchev, 2003] Introduction Orbit matrices of symmetric Let D be a 2-( v , k , λ ) design with a fixed-point-free and designs Codes from fixed-block-free automorphism φ of order q , where q is orbit matrices prime. Further, let M be the orbit matrix induced by the action of symmetric designs of the group G = � φ � on the design D . If p is a prime dividing Self-dual r and λ then the orbit matrix M generates a self-orthogonal codes from extended orbit code of length b | q over F p . matrices 8 / 29
Orbit matrices Let a group G acts on a symmetric ( v , k , λ ) design with t = v of symmetric Ω designs and orbits of length Ω on the set of points and set of blocks. related self-dual codes Theorem 1a Introduction Orbit matrices Let D be a symmetric ( v , k , λ ) design admitting an of symmetric designs automorphism group G that acts on the sets of points and Codes from orbit matrices blocks with t = v Ω orbits of length Ω. Further, let M be the of symmetric designs orbit matrix induced by the action of the group G on the Self-dual design D . If p is a prime dividing k and λ , then the rows of the codes from extended orbit matrix M span a self-orthogonal code of length t over F p . matrices 9 / 29
Self-dual codes from extended orbit matrices Orbit matrices In the sequel we will study codes spanned by orbit matrices for of symmetric designs and a symmetric ( v , k , λ ) design and orbit lengths distribution related self-dual codes (Ω , . . . , Ω), where Ω = v t . We follow the ideas presented in: Introduction • E. Lander, Symmetric designs: an algebraic approach, Orbit matrices of symmetric Cambridge University Press, Cambridge (1983). designs Codes from • R. M. Wilson, Codes and modules associated with designs orbit matrices of symmetric and t -uniform hypergraphs, in: D. Crnkovi´ c, V. Tonchev, designs (eds.) Information security, coding theory and related Self-dual codes from combinatorics, pp. 404–436. NATO Sci. Peace Secur. extended orbit matrices Ser. D Inf. Commun. Secur. 29 IOS, Amsterdam (2011). (Lander and Wilson have considered codes from incidence matrices of symmetric designs.) 10 / 29
Orbit matrices of symmetric designs and related Theorem 2 self-dual codes Let p be a prime. Suppose that C is the code over F p spanned Introduction by the incidence matrix of a symmetric ( v , k , λ ) design. Orbit matrices of symmetric designs 1 If p | ( k − λ ), then dim ( C ) ≤ 1 2 ( v + 1). Codes from orbit matrices 2 If p ∤ ( k − λ ) and p | k , then dim ( C ) = v − 1. of symmetric designs 3 If p ∤ ( k − λ ) and p ∤ k , then dim ( C ) = v . Self-dual codes from extended orbit matrices 11 / 29
Orbit matrices Theorem 3 [D. Crnkovi´ c, SR] of symmetric designs and related self-dual codes Let a group G acts on a symmetric ( v , k , λ ) design D with t = v Ω orbits of length Ω, on the set of points and the set of Introduction Orbit matrices blocks, and let M be an orbit matrix of D induced by the of symmetric designs action of G . Let p be a prime. Suppose that C is the code Codes from orbit matrices over F p spanned by the rows of M . of symmetric designs 1 If p | ( k − λ ), then dim ( C ) ≤ 1 2 ( t + 1). Self-dual codes from 2 If p ∤ ( k − λ ) and p | k , then dim ( C ) = t − 1. extended orbit matrices 3 If p ∤ ( k − λ ) and p ∤ k , then dim ( C ) = t . 12 / 29
Orbit matrices Let a group G acts on a symmetric ( v , k , λ ) design with t = v of symmetric Ω designs and orbits of length Ω on the set of points and set of blocks. related self-dual codes Theorem 1a Introduction Orbit matrices Let D be a symmetric ( v , k , λ ) design admitting an of symmetric designs automorphism group G that acts on the sets of points and Codes from orbit matrices blocks with t = v Ω orbits of length Ω. Further, let M be the of symmetric designs orbit matrix induced by the action of the group G on the Self-dual design D . If p is a prime dividing k and λ , then the rows of the codes from extended orbit matrix M span a self-orthogonal code of length t over F p . matrices 13 / 29
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