Linear Algebra and its Applications

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1 Linear Algebra and its Applications 434 (2011) Contents lists available at ScienceDirect Linear Algebra and its Applications journal homepage: A generalization of the formulas for intersection numbers of dual polar association schemes and their applications Jianmin Ma a, Jun Guo b,, Fenggao Li c,kaishunwang d a Oxford College of Emory University, Oxford, GA 30054, USA b Math. and Inf. College, Langfang Teachers College, Langfang , China c Dept. of Math., Hunan Institute of Science and Technology, Yueyang , China d Sch. Math. Sci. & Lab. Math. Com. Sys., Beijing Normal University, Beijing , China ARTICLE INFO Article history: Received 18 March 2010 Accepted 4 November 2010 Availableonline4December2010 Submitted by R.A. Brualdi AMS classification: 20G40 05E30 20B25 94A62 05C25 ABSTRACT Dual polar association schemes form an important family of association schemes, whose intersection numbers were computed in [Wan et al., Studies in Finite Geometry and the Construction of Incomplete Block Designs, Science Press, Beijing, 1966 (in Chinese)]. In this paper, we generalize the formulas for the intersection numbers, and introduce their applications to pooling designs, Cartesian authentication codes and vertex-transitive graphs Published by Elsevier Inc. Keywords: Dual polar scheme Totally isotropic subspace Pooling design Cartesian authentication code Vertex transitive graph 1. Introduction In this section we shall introduce the classical spaces, following notation and terminology from Wan s book [19]. The classical spaces associated with the three types of forms (alternating bilinear, hermitian, and uadratic) are referred to by the same names as the groups associated with them: Corresponding author. addresses: Jianmin.Ma@gmail.com (J. Ma), guojun lf@163.com (J. Guo), fenggaol@163.com (F. Li), wangks@bnu.edu.cn (K. Wang) /$ - see front matter 2010 Published by Elsevier Inc. doi: /j.laa

2 J. Ma et al. / Linear Algebra and its Applications 434 (2011) symplectic, unitary, and orthogonal respectively. In this paper, we are concerned with those whose underlying vector spaces are of even dimension over a finite field. Throughout this paper, we denote by F a finite field with elements and by F 2ν the 2ν-dimensional row vector over F for a fixed a positive integer ν.foranm-dimensional subspace P in F 2ν,wemean by a matrix representation of P an m 2ν matrix whose rows form a basis of P, denotedbythesame symbol P. We shall work with partitioned matrices whose entries are themselves submatrices. For typographical convenience, we sometimes leave blank the zero submatrices. We write I (r) for the identity matrix of size r, and we omit r if it is clear from the context. Let K = 0 I(ν) I (ν). 0 The symplectic group of degree 2ν over F with respect to K, denoted by Sp 2ν (F ), consists of all 2ν 2ν matrices T over F satisfying TKT t = K, where T t denotes the transpose of T. The space F 2ν together with the right multiplication action of Sp 2ν (F ) is called the 2ν-dimensional symplectic space over F.Anm-dimensional subspace P in F 2ν is said to be of type (m, s) if PKP t is of rank 2s. Inparticular, subspacesoftype (m, 0) are called m-dimensional totally isotropic subspaces and ν-dimensional totally isotropic subspaces are called maximal totally isotropic subspaces. Let = p 2, where p is a prime power. Then F has an involutive automorphism a a = a p. Let H = 0 I(ν) I (ν). 0 The unitary group of degree 2ν over F, denoted by U 2ν (F ), consists of all 2ν 2ν matrices T over F satisfying THT t = H. The vector space F 2ν together with the right multiplication action of U 2ν(F ) is called the 2ν-dimensional unitary space over F.Anm-dimensional subspace P in F 2ν is said to be of type (m, r) if PKP t is of rank r. Similarly, subspaces of type (m, 0) are called m-dimensional totally isotropic subspaces and ν-dimensional totally isotropic subspaces are called maximal totally isotropic subspaces. It is more involved to define orthogonal spaces. Denote by K 2ν the set of all 2ν 2ν alternate matrices over F.Two2ν 2ν matrices A and B over F are said to be congruent mod K 2ν,denoted by A B (mod K 2ν ), if A B K 2ν.Clearly, is an euivalence relation on the set of all 2ν 2ν matrices. Let [A] denote the euivalence class containing A. Two matrix classes [A] and [B] are said to be cogredient if there is a nonsingular 2ν 2ν matrix Q over F such that [QAQ t ] [B]. Let S 2ν = 0 I(ν) or 0 I(ν) I (ν) 0 0 according the being odd or even, respectively. The orthogonal group of degree 2ν over F with respect to S 2ν,denotedbyO 2ν (F ), consists of all 2ν 2ν matrices T over F satisfying [TS 2ν T t ] [S 2ν ]. The space F 2ν together with the right multiplication action of O 2ν (F ) is called the 2ν-dimensional orthogonal space over F.Forbeing odd, let 0 I (s) S 2s+γ,Γ = I (s) 0, if γ = 0,, Γ = (1) or (z), if γ = 1, Γ diag(1, z), if γ = 2,

3 1274 J. Ma et al. / Linear Algebra and its Applications 434 (2011) where z is a fixed non-suare element of F.For being even, let 0 I (s) S 2s+γ,Γ = 0, Γ = Γ, if γ = 0, (1), if γ = 1, α 1, if γ = 2, α where α is a fixed element of F such that α / {x 2 + x x F }.Anm-dimensional subspace P is a subspace of type (m, 2s + γ,s, Γ ) if PS 2ν P t is cogredient to diag (S 2s+γ,Γ, 0 (m 2s γ) ).IfΓ is the empty set, we omit the symbol. In particular, subspaces of type (m, 0, 0) are called m-dimensional totally isotropic subspaces and ν-dimensional totally isotropic subspaces are called maximal totally isotropic subspaces. Let G 2ν be one of the three classical groups acting on F 2ν.By[19, Theorems 3.7, 5.8, 6.4 and 7.6], the set of subspaces of the same type in F 2ν forms an orbit under G 2ν.DenotebyM(m, 0; 2ν) the set of all m-dimensional totally isotropic subspaces of F 2ν. The action of G 2ν on M(ν, 0; 2ν) gives rise to a dual polar association scheme (see [1]). Wan et al. [20] computed all intersection numbers of these dual polar schemes. As a generalization of dual polar schemes, we constructed association schemes from singular classical spaces (see [6 9]). For 1 m ν, suppose P, Q M(ν, 0; 2ν), U M(m, 0; 2ν) satisfying dim(p Q) = dim(p U) = i. Let P i jk (ν, ν; m) = {R M(m, 0; 2ν) dim(p R) = j, dim(q R) = k}, P i jk (ν, m; ν) = {R M(ν, 0; 2ν) dim(p R) = j, dim(u R) = k}. Note that the size of P i jk (ν, ν; ν) is an intersection numbers of the dual polar scheme based on M(ν, 0; 2ν). In this paper, we focus on the sizes of P i (ν, ν; m) jk and Pi jk (ν, m; ν), and discuss their applications. In Section 2 we calculate the sizes of these two sets. In Section 3 we construct a family of error-correcting pooling designs, and exhibit its disjunct property. We construct a family of Cartesian authentication codes and vertex transitive graphs in Sections 4 or 5, respectively. 2. The cardinalities of P i (ν, ν; m) jk and Pi jk (ν, m; ν) In this section we shall compute the sizes of P i (ν, ν; m) jk and Pi jk (ν, m; ν).webeginwithsome useful lemmas. Lemma 2.1 [19, Theorems 3.30, 4.9, 5.31 and 6.37]. Let S(t, m), H(t, m) and K(t, m) be the sets of all m m symmetric, hermitian and alternate matrices of rank t over F, respectively. Then S(t, m) = t/2 ( t/2 +1) H(t, m) = t(t 1)/4 K(t, m) = t/2(t/2 1) m l= t/2 +1 m i=m t+1 m l=m t+1 ( l 1) / t/2 m t ( l + 1) ( l 1), l=1 ( i 1) / t ( i/2 ( 1) i ), i=1 ( l 1) / t/2 ( 2l 1). l=1 l=1

4 J. Ma et al. / Linear Algebra and its Applications 434 (2011) Lemma 2.2 [20, Chapter 1, Theorem 5]. The number of m n matrices with rank i over F is where N(i; m n) = i(i 1)/2 [ m i ] n t=n i+1 ( t 1), [ ] m = m i l=m i+1 (l 1) / i l=1 (l 1). Lemma 2.3. Let 1 m n. Then the number of all m nmatrices(ab) of rank r over F,withA respectively symmetric, hermitian and alternate, is given by N (r; m n) = r t=r min{n m,r} S(t, m), in the symmetric case, t(n m) N(r t; (m t) (n m)) H(t, m), in the hermitan case, K(t, m), in the alternate case. Proof. For any m m matrix A of rank t,letm(a) denote the set of all m n matrices (A B) of rank r over F.Foreach(A B) M(A), there exists a nonsingular m m matrix T such that T(A B) = (TA TB) = ( m n m ) A 1 B 1 t, 0 B 2 m t and thus we have rank B 2 = r t, M(A) = M(TA). By Lemma 2.2, M(A) = M(TA) = t(n m) N(r t; (m t) (n m)). Since r min{n m, r} rank A r, the desired formula follows from Lemma 2.1. Lemma 2.4. Let 1 m ν,p 1, P 2 M(ν, 0; 2ν) and Q 1, Q 2 M(m, 0; 2ν).Thendim(P 1 Q 1 ) = dim(p 2 Q 2 ) if and only if there exists a T G 2ν such that P 1 T = P 2, Q 1 T = Q 2. Proof. We illustrate with the symplectic case here. Suppose that P 1 Q 1 = D 1, P 2 Q 2 = D 2 and dim(p 1 Q 1 ) = dim(p 2 Q 2 ) = i. ThenP 1, Q 1, P 2, Q 2 have matrix representations of the forms P 1 = P 11, Q 1 = D 1, P 2 = P 21, Q 2 = D 2, D 1 Q 11 D 2 Q 21 respectively. Since dim(p 1 + Q 1 ) = ν + m i, by[19, Theorem 3.6], rank (P 11 KQ11 t ) m i, which implies that rank (P 11 KQ11 t ) = m i. Similarly, rank (P 21KQ21 t ) = m i. Hence,thereexisttwo (ν i) (ν i) nonsingular matrices A 1, A 2 and two (m i) (m i) nonsingular matrices B 1, B 2

5 1276 J. Ma et al. / Linear Algebra and its Applications 434 (2011) such that t t A 1 P 11 A 1 P 11 A 2 P 21 A 2 P I(m i) D 1 K D 1 = D 2 K 0 D 2 = B 1 Q 11 B 1 Q 11 B 2 Q 21 B 2 Q 21 (I (m i) 0) 0 0 By [19, Theorem 3.11], there exists a T Sp 2ν (F ) such that P 1 T = P 2, Q 1 T = Q 2. The converse is obvious. For 1 m ν, suppose that P, Q M(ν, 0; 2ν) with dim(p Q) = i. Recall that P i jk (ν, ν; m) is the set of all R M(m, 0; 2ν) satisfying dim(p R) = j and dim(r Q) = k. By Lemma 2.4, p i (ν, ν; m) jk = Pi jk (ν, ν; m) is independent of any particular choices of P and Q with dim(p Q) = i. Theorem 2.5. Let 1 m ν. Thenp i jk (ν, ν; m) = α+γ ν i,β+ρ i,β+γ =k α+β+γ +ρ=m,β j α+β α+β j min{α+β j,ν i γ α} t α+β j N α [ ] [ ] [ ] [ ] ν (α+γ)(i β ρ)+ρ(2ν i m) i i ν i α i β α β γ ρ (α + β j; α (ν i γ)) ρ2 /2, ρ(ρ+1)/2, the symplectic case, the unitary case, ρ(ρ 1)/2, the orthogonal case. Proof. By Lemma 2.4, wemaytake P = (I (ν) 0 (ν) ), Q = (0 (ν,ν i) I (ν) 0 (ν,i) ). For any R P i jk (ν, ν; m), writer in block form R = ( ν i i ν i i R 1 R 2 R 3 R 4 ). Suppose that rank R 4 = ρ, rank (R 1 R 4 ) = ρ + α, rank (R 1 R 3 R 4 ) = ρ + α + γ and β = m (ρ + α + γ). By suitable row elementary transformations, R may be reduced to the form ν i i ν i i R 11 R 12 R 13 0 α 0 R β 0 R 32 R 33 0, γ R 41 R 42 R 43 R 44 ρ [ ] ν i α where rank R 44 = ρ, rank R 11 = α, rank R 33 = γ and rank R 22 = β. Notethatthereare [ i choices for subspace R 11 and β] choices for R 22. By the transitivity of G 2ν on the set of subspaces with

6 J. Ma et al. / Linear Algebra and its Applications 434 (2011) the same type, the number of R does not depend on any particular choices for R 11 and R 22.Without loss of generality we may take R 11 = (I (α) 0), R 22 = (I (β) 0). Then R has a matrix representation of the form α ν i α β i β α ν i α β i β I 0 0 R 122 R 131 R α 0 0 I β R 322 R 331 R (1) γ 0 R R 422 R 431 R 432 R 441 R 442 ρ Since R is totally isotropic, we have R 331 = [ 0 and R ] 441 = 0. So rank R 332 = γ,rank [ R 442 ] = ρ and γ ν i α, ρ ν i α i β.notethatthereare γ choices for subspace R i β 332 and ρ choices for R 442. Similarly, the number of R does not depend on any particular choices for R 332 and R 442.Without loss of generality we may take R 332 = (I (γ ) 0), R 442 = (I (ρ) 0). Then R must have the following uniue matrix representation α γ ν i α γ β ρ i β ρ α γ ν i α γ β ρ i β ρ I R 1221 R 1222 R R α I β R 3221 R I (2) γ 0 R 4121 R R 4221 R 4222 R R I 0 ρ For the symplectic case, R131 t = R 131, R 1322 R4122 t = Rt R 1221, R4121 t = R 3221, R 4122 R4322 t R 4322R4122 t = Rt 4221 R 4221; for the unitary case, R R t = 0, 131 R 1322R t Rt R 1221 = 0, R t R 3221 = 0, R 4122 R t R 4322R t Rt R 4221 = 0; for the odd orthogonal case (i.e., the base field F is of odd characteristic), R R131 t = 0, R 1322R4122 t + Rt R 1221 = 0, R4121 t + R 3221 = 0, R 4122 R4322 t + R 4322R4122 t + Rt R 4221 = 0; (3) (4) (5) for the even orthogonal case, R 131 0, R 1322 R t Rt R , R t R 3221 = 0, R 4122 R t R 4221 = 0. (6) From dim(p R) = j and dim(q R) = k we deduce that j α + β, β + γ = k and rank (R 131 R 1322 ) = α + β j. By Lemma 2.3 and (3) (6), the number of matrices (R 131 R 1322 ) of rank α + β j is

7 1278 J. Ma et al. / Linear Algebra and its Applications 434 (2011) Nα (α + β j; α (ν i γ )). Once R 131, R 1322, R 4122, R 4322, R 431 and R 4121 are fixed, the matrices R 1221 and R 3221 are uniuely determined; and there are ρ(ρ+1)/2, ρ2 /2, ρ(ρ 1)/2, ρ(ρ 1)/2 choices for R 4221 such that (3), (4), (5) and (6) hold, respectively. Since R 1222, R 3222 and R 4222 may be any α (i β ρ), γ (i β ρ) and ρ (i β ρ) matrices over F, respectively, there are (i β ρ)(α+γ +ρ) choices for R 1222, R 3222 and R Therefore, the number of subspaces of the form (2)is ρ(ρ+1)/2, the symplectic case, (α+γ)(i β ρ)+ρ(2ν i m) Nα (α + β j; α (ν i γ)) ρ2 /2, the unitary case, ρ(ρ 1)/2, the orthogonal case. Hence the desired assertion follows. Corollary 2.6. Let 1 m ν.then p ν jj (ν, ν; m) = (m j)(ν m) [ ν j ] [ ] ν j m j (m j)(m j+1)/2, the symplectic case, (m j)2 /2, the unitary case, (m j)(m j 1)/2, the orthogonal case. For 1 m ν, letp M(ν, 0; 2ν), U M(m, 0; 2ν) with dim(p U) = i. Recall that P i jk (ν, m; ν) consists of R M(ν, 0; 2ν) with dim(p R) = j and dim(u R) = k. By Lemma 2.4, p i (ν, m; ν) jk = Pi jk (ν, m; ν) is independent of any particular choice for P and U with dim(p U) = i. Corollary 2.7. We have p i (ν, m; ν) = jk pj (ν, ν; m) ik pν(ν, ν; jj ν)/ p ν ii (ν, ν; m). Proof. Let M ={(P, Q, R) P, R M(ν, 0; 2ν), Q M(m, 0; 2ν), dim(p Q) = i, dim(p R) = j, dim(q R) = k}. We count M in two different ways. For a fixed subspace P M(ν, 0; 2ν), by Corollary 2.6, there are p ν ii (ν, ν; m) subspaces Q M(m, 0; 2ν) satisfying dim(p Q) = i. For two fixed subspaces P M(ν, 0; 2ν), Q M(m, 0; 2ν) with dim(p Q) = i, there are p i jk (ν, m; ν) subspaces R M(ν, 0; 2ν) satisfying dim(p R) = j and dim(q R) = k. It follows that M =p i (ν, m; ν) jk pν ii (ν, ν; m) N(ν, 0; 2ν). For a fixed subspace P M(ν, 0; 2ν), by Corollary 2.6 again, there are p ν jj (ν, ν; ν) subspaces R M(ν, 0; 2ν) satisfying dim(p R) = j. For two fixed maximal totally isotropic subspaces P, R with dim(p R) = j, there are p j ik (ν, ν; m) subspaces Q M(m, 0; 2ν) satisfying dim(p Q) = i and dim(r Q) = k.bytheorem2.5, M =p j (ν, ν; m) ik pν jj (ν, ν; ν) N(ν, 0; 2ν). Hence the desired assertion follows.

8 J. Ma et al. / Linear Algebra and its Applications 434 (2011) Remark. The action of G 2ν on M(m, 0; 2ν) M(v, 0; 2ν) defines a coherent configuration (see [10] for coherent configurations). The parameters p i (ν, ν; m) jk and pi jk (ν, m; ν) are intersection numbers of this configuration. 3. Pooling designs A binary matrix is d e -disjunct if for any column C and any d others columns, there exist at least e rows such that each of them has value 1 at column C and value 0 at all the other d columns. A d 1 -disjunct matrix is also called d-disjunct. d e -disjunct matrices form the basis for error-tolerant nonadaptive group testing algorithms. These algorithms have applications in many areas such as DNA library screening [2]. There are several constructions of d e -disjunct matrices in the literatures [3,5,12,14 16]. In this section, as an application of the theorems in Section 2, we shall construct a family of d e - disjunct matrices and discuss their disjunct property. For 1 m <νand 0 j m, letm be the matrix with rows and columns respectively indexed by M(m, 0; 2ν) and M(ν, 0; 2ν) in which M(P, Q) = 1ifdim(P Q) = j and 0 otherwise. Lemma 3.1 [19, Corollaries 3.19, 5.20, 6.23 and 7.25]. Let 1 m ν. Then the number of m-dimensional is totally isotropic subspaces in F 2ν ν ( 2i 1) / m ( i 1), the symplectic case, i=ν m+1 i=1 ν N(m, 0; 2ν) = ( i 1)( i 1/2 + 1) / m ( i 1), the unitary case, i=ν m+1 i=1 ν ( i 1)( i 1 + 1) / m ( i 1), the orthogonal case. i=ν m+1 i=1 Lemma 3.2. Let 1 m ν and P M(m, 0; 2ν). Then the number of subspaces Q M(ν, 0; 2ν) with dim(p Q) = jisn(ν, 0; 2ν)p ν jj (ν, ν; m)/ N(m, 0; 2ν). Proof. Let M ={(P, Q) P M(m, 0; 2ν), Q M(ν, 0; 2ν), dim(p Q) = j}. Now we count M in two different ways. For a fixed subspace P of type (m, 0), suppose that there are α maximal totally isotropic subspaces Q satisfying dim(p Q) = j. By Lemma 3.1 M =α N(m, 0; 2ν). For a fixed maximal totally isotropic subspace Q, there are p ν jj (ν, ν; m) subspaces P satisfying dim(p Q) = j. By Lemma 3.1 and Corollary 2.6, M =p ν jj (ν, ν; m)n(ν, 0; 2ν). Hence α = N(ν, 0; 2ν)p ν jj (ν, ν; m)/ N(m, 0; 2ν), as desired. By Lemmas 3.1, 3.2 and Corollary 2.6, M is an N(m, 0; 2ν) N(ν, 0; 2ν)matrix, which has constant row weight N(ν, 0; 2ν)p ν (ν, ν; m)/n(m, 0; 2ν) jj and constant column weight pν jj (ν, ν; m). Theorem 3.3. Let 1 m <νand 0 j m. If 1 d p ν (ν, ν; jj m)/α + 1,thenMisd e -disjunct, where e = p ν (ν, ν; m) jj dα, α = max{pl jj (ν, ν; m) 0 l ν 1}.

9 1280 J. Ma et al. / Linear Algebra and its Applications 434 (2011) Proof. Pick any d + 1 distinct columns C, C 1, C 2,..., C d of M.ByTheorem2.5, we may assume that the number of subspaces P M(m, 0; 2ν) satisfying dim(p C) = j and dim(p C i ) = j is at most α = max{p l jj (ν, ν; m) 0 l ν 1}. Hence the number of subspaces P of M(m, 0; 2ν) satisfying dim(p C) = j and dim(p C 1 ),..., dim(p C d ) = j is at least e = p ν jj (ν, ν; m) dα, from which follows that M is d e -disjunct. Since e 1, we obtain p ν jj (ν, ν; m) d + 1, α as desired. Remark. If j = m, the disjunct matrices in Theorem 3.3 are the disjunct matrices based on pooling spaces in [11, Example 4.2]. 4. Authentication codes Authentication codes were invented in 1974 by Gilbert et al. [4] for protecting the integrity of information. For a survey of authentication codes, we recommend Simmons [17]. In 1992, Wan [18] constructed Cartesian authentication codes from the unitary space. In this section we shall construct a family of authentication codes, following notation and terminology in [18]. As a by-product, we obtain some enumeration formulas in 2ν-dimensional classical spaces. Theorem 4.1. Let F 2ν be a 2ν-dimensional classical spaces. Suppose that 2 i ν 1 and P 0 = (I (ν) 0 (ν) ). Define the source states to be all subspaces of dimension i contained in P 0, the encoding rules to be the maximal totally isotropic subspaces intersecting P 0 at {0}, andthemessagestobethesubspaces of type ϑ intersecting P 0 at i-dimensional subspaces, where ϑ = (ν + i, i), (ν + i, 2i) or (ν + i, 2i, i) according to the symplectic, unitary or orthogonal case, respectively. Denote the set of source states, the set of encoding rules, and the set of messages by S, E and M, respectively. Given any P S and any P 1 E, P + P 1 is a message into which the source state P is encoded under the encoding rule P 1.Theabove construction yields a Cartesian authentication code, whose parameters are [ ] ν S =, i E = ν2 /2, M = ν(ν+1)/2, the symplectic case, the unitary case, ν(ν 1)/2, the orthogonal case, [ ] (ν i)(ν+i+1)/2 ν, the symplectic case, i [ ] (ν i)(ν+i)/2 ν, the unitary case, i [ ] (ν i)(ν+i 1)/2 ν, the orthogonal case. i

10 J. Ma et al. / Linear Algebra and its Applications 434 (2011) Suppose that the encoding rule is chosen according to a uniform probability distribution, and denote the probabilities of a successful impersonation attack and a successful substitution attack by P I and P S, respectively. Then 1, the symplectic case, (ν i)(ν+i+1)/2 1 P I =, (ν i)(ν+i)/2 the unitary case, 1, the orthogonal case, (ν i)(ν+i 1)/2 1, i the symplectic case, 1 P S = and P I is optimal., the unitary case, i 1/2 1 i 1, the orthogonal case, Proof. Let P be a source state and P 1 be an encoding rule. Since P P 0 and P 0 P 1 ={0}, P + P 1 is of type ϑ. By dim((p + P 1 ) P 0 ) = dim(p + P 1 ) + dim P 0 dim((p + P 1 ) + P 0 ) = i, we have (P + P 1 ) P 0 = P. It follows that P + P 1 is a message. So P 1 defines a map f : S M by f (P) = P + P 1. Next, let Q be a message such that P = P 0 Q. Then P isasourcestate.wemaytake Q = ( ν ν ) B 0 i A I ν, where P = (B 0). Since rank B = i, that there exists a ν ν nonsingular matrix T 1 such that BT 1 = (I (i) 0 (i,ν i) ).Let T = diag (T 1, S 1 1 ) G 2ν, where S 1 = T1 t, T t 1 or T 1 t according to the symplectic, unitary or orthogonal case, respectively, such that P 0 T = P 0 and QT = i ν i i ν i I A 1 I 0 i i. 0 A 2 0 I ν i Since QT is of type ϑ,wehavea 2 A t = 0, 2 A 2 + A t = 2 0orA 2 + A t 2 = 0 according to the symplectic, unitary or orthogonal case, respectively. Take P 1 T 1 = ( i ν i i ν i 0 A 1 I 0 B 1 A 2 0 I ) i ν i, where B 1 = A t, 1 At 1 or At 1 according to the symplectic, unitary or orthogonal case, respectively. Then P 1 E satisfying P + P 1 = Q. Therefore, f is a surjective map.

11 1282 J. Ma et al. / Linear Algebra and its Applications 434 (2011) Suppose that there is another source state P encoded into the message Q.ThenP P 0 and P Q; and so P P 0 Q = P, which implies that P = P.HencethesourcestateP is uniuely determined by Q. Clearly, S = [ ] ν i. By Corollary 2.6, E =pν 00 (ν, ν; ν). Let Q be a message. Without loss of generality, we may take Q = I(i) 0 (i,ν i) 0 (i,ν) 0 0 I (ν). Then the encoding rules contained in Q have a matrix representations of the form A 1 0 (ν,ν i) I (i) 0 A I (ν i). (7) The subspaces of the form (7) aretotallyisotropic,soa 2 = 0 and A 1 A t = 0, 1 A 1 + A t = 1 0or A 2 + A t 2 = 0 according to the symplectic, unitary or orthogonal case, respectively. Hence, the number of encoding rules contained in Q is α = i(i+1)/2, i2 /2 or i(i 1)/2 according to the symplectic, unitary or orthogonal case, respectively. So M = S E /α. Let Q, Q be two distinct messages containing a common encoding rule P 1, and let P, P be the uniue source states contained in them, respectively. Then P = Q P 0, P = Q P 0 and Q = P P 1, Q = P P 1, where X Y denotes the direct sum of X and Y. We claim that Q Q = (P P ) P 1.Foranyw Q Q,wehavew = x + z 1 = x + z 2, where x P, x P and z 1, z 2 P 1.SinceP + P P 0, x x = z 2 z 1 P 0 P 1, which implies z 1 = z 2 and x = x.thenw (P P ) P 1, and we prove the claim. Let dim(p P ) = r. Since Q = Q,wehavemax{2i m, 0} r i 1. We assert that the set of encoding rules contained in both Q and Q coincides with the set of maximal totally isotropic subspaces P 1 contained in Q Q such that P 1 (P P ) ={0}. Indeed, let P 1 be the maximal totally isotropic subspace contained in Q Q such that P 1 (P P ) ={0}. Then P 1 P 0 Q Q P 0 Q P 0, Q P 0. Therefore, P 1 P 0 P 1 (P P ) ={0}, which implies that P 1 is an encoding rule contained in both Q and Q. Conversely, let P 1 be an encoding rule contained in both Q and Q, that is, P 1 is a maximal totally isotropic subspace contained in both Q and Q such that P 1 P ={0} and P 1 P ={0}.Then P 1 is a maximal totally isotropic subspace contained in Q Q such that P 1 (P P ) ={0}, and we prove the above assertion. Hence the number of encoding rules contained in Q and Q is r(r+1)/2, r2 /2 or r(r 1)/2 according to the symplectic, unitary or orthogonal case, respectively. Since max{2i m, 0} r i 1, we obtain P I and P S. It is obvious that P I is optimal. Corollary 4.2. Let F 2ν totally isotropic subspace of F 2ν be a 2ν-dimensional classical space. Let 0 i ν and P 0 be a fixed maximal. Then the number of subspaces Q of type ϑ satisfying dim(p 0 Q) = iis [ ] (ν i)(ν+i+1)/2 ν, the symplectic case, i n i (ν; ϑ) = [ ] (ν i)(ν+i)/2 ν, the unitary case, i [ ] (ν i)(ν+i 1)/2 ν, the orthogonal case, i

12 J. Ma et al. / Linear Algebra and its Applications 434 (2011) where ϑ = (ν + i, i), (ν + i, 2i) or (ν + i, 2i, i) according to the symplectic, unitary or orthogonal case, respectively. Corollary 4.3. For 0 i ν,letq 0 beafixedsubspaceoftypeϑ in F 2ν,whereϑ = (ν +i, i), (ν +i, 2i) or (ν + i, 2i, i) according to the symplectic, unitary or orthogonal case, respectively. Then the number of maximal totally isotropic subspaces P satisfying dim(p Q 0 ) = iis n i (ν; ϑ)n(ν, 0; 2ν), N(ϑ; 2ν) where N(ϑ; 2ν) is the number of subspaces of F 2ν of type ϑ (see [19]). Proof. The proof is similar to that of Lemma 3.2, and thus omitted. 5. Graphs In this section we shall construct a family of vertex transitive graphs. For 0 i m ν, suppose that W 0 is a fixed m-dimensional totally isotropic subspace in F 2ν. Let X (i) be the set of all the maximal totally isotropic subspaces P of F 2ν satisfying dim(p W 0 ) = i. Define a graph Γ whose vertex set is the set X (i), and two vertices P and Q are adjacent if and only if dim(p Q) = ν 1. Then Γ is a subgraph of the dual polar graph based on M(ν, 0; 2ν).Ifm = ν, then Γ is the (ν i)-th subconstituent of (see [13,21,22]). The stabilizer G 2ν (W 0 ) of W 0 in G 2ν is an automorphism group of Γ, by Lemma 2.4 and Corollary 2.7, Γ is a vertex transitive graph of degree p i ν 1,i (ν, m; ν). Lemma 3.2 implies that Γ has N(ν, 0; 2ν)p ν(ν, ν; ii m)/ N(m, 0; 2ν) vertices. Let G 2ν (W 0 ) act on the set X (i) X (i) in a natural way as (P, Q)T = (PT, QT), P, Q X (i), T G 2ν (W 0 ). Let Λ 0, Λ 1,...,Λ t be the orbits of this action. Since G 2ν (W 0 ) acts transitively on X (i), the configuration (X (i), {Λ j } 0 j t ) forms a symmetric association scheme (see [1]). We will study these association schemes in a separate paper. Acknowledgment We would like thank the referees for their valuable suggestions. This research is supported by NCET , NSF of China ( , ), Hunan Provincial Natural Science Foundation of China (09JJ3006), and the Fundamental Research Funds for the Central Universities of China. References [1] E. Bannai, T. Ito, Algebraic Combinatorics I: Association Schemes, The Benjamings/Cummings Publishing Company, Inc., [2] D. Du, F. Hwang, Pooling Designs and Non-adaptive Group Testing: Important Tools for DNA Seuencing, World Scientific, [3] A.G. D yachkov, F.K. Hwang, A.J. Macula, P.A. Vilenkin, C. Weng, A construction of pooling designs with some happy surprises, J. Comput. Biol. 12 (2005) [4] E.N. Gilbert, F.J. MacWilliams, N.J.A. Sloane, Codes which detect deception, Bell Syst. Tech. J. 53 (1974) [5] J. Guo, Pooling designs associated with unitary space and ratio efficiency comparison, J. Comb. Optim. 19 (2010) [6] J. Guo, Suborbits of (m, k)-isotropic subspaces under finite singular classical groups, Finite Fields Appl. 16 (2010) [7] J. Guo, K. Wang, Suborbits of m-dimensional totally isotropic subspaces under finite singular classical groups, Linear Algebra Appl. 430 (2009) [8] J. Guo, K. Wang, F. Li, Association schemes based on maximal isotropic subspaces in singular classical spaces, Linear Algebra Appl. 430 (2009) [9] J. Guo, K. Wang, F. Li, Association schemes based on maximal isotropic subspaces in singular pseudo-symplectic spaces, Linear Algebra Appl. 431 (2009)

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