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Codes and Modules Associated with Designs and T-Uniform Hypergraphs Richard M

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Codes and Modules Associated with Designs and T-Uniform Hypergraphs Richard M Codes and modules associated with designs and t-uniform hypergraphs Richard M. Wilson California Institute of Technology (1) Smith and diagonal form (2) Solutions of linear equations in integers (3) Square incidence matrices (4) A chain of codes (5) Self-dual codes; Witt’s theorem (6) Symmetric and quasi-symmetric designs (7) The matrices of t-subsets versus k-subsets, or t-uniform hy- pergaphs (8) Null designs (trades) (9) A diagonal form for Nt (10) A zero-sum Ramsey-type problem (11) Diagonal forms for matrices arising from simple graphs 1. Smith and diagonal form Given an r by m integer matrix A, there exist unimodular matrices E and F ,ofordersr and m,sothatEAF = D where D is an r by m diagonal matrix. Here ‘diagonal’ means that the (i, j)-entry of D is 0 unless i = j; but D is not necessarily square. We call any matrix D that arises in this way a diagonal form for A. As a simple example, ⎛ ⎞ 0 −13 10 314⎜ ⎟ 100 ⎝1 −1 −1⎠ = . 21 4 −27 050 01−2 The matrix on the right is a diagonal form for the middle matrix on the left. Let the diagonal entries of D be d1,d2,d3,.... ⎛ ⎞ 20 0 ··· ⎜ ⎟ ⎜0240 ···⎟ ⎜ ⎟ ⎝0 0 120 ···⎠ . ... If all diagonal entries di are nonnegative and di divides di+1 for i =1, 2,...,thenD is called the integer Smith normal form of A, or simply the Smith form of A, and the integers di are called the invariant factors,ortheelementary divisors of A.TheSmith form is unique; the unimodular matrices E and F are not. As we have defined them, the number of invariant factors of a matrix (or the number of diagonal entries of a diagonal form) is equal to the minimum of the number of rows and the number of columns. But here and in the sequel, di may be interpreted as 0 if the index i exceeds the number of rows or columns. It is clear that the invariant factors (or diagonal entries) of A and A are the same apart from trailing zeros 0. Some examples follow. 314→ 314→ 311 4 −27 1 −33 1 −32 → 301→ 001→ 001 1 −52 −552 052 → 001 → 100 050 050 ⎛ ⎞ 2 853624966054709580 ⎜ ⎟ ⎜ ⎟ ⎜50 46 82 88 94 25 2 1 21 40⎟ ⎜ ⎟ ⎜8586524 244557943858⎟ ⎜ ⎟ ⎜95 89 31 49 23 1 74 21 69 81⎟ ⎜ ⎟ ⎜18 57 28 27 39 21 70 45 38 89⎟ ⎜ ⎟ ⎜ ⎟ ⎜22 97 86 90 78 97 42 74 69 30⎟ ⎜ ⎟ ⎜53 63 63 31 23 88 38 56 61 28⎟ ⎜ ⎟ ⎜98 61 95 16 23 68 32 6 78 17⎟ ⎜ ⎟ ⎝47 81 42 41 59 68 18 16 37 73⎠ 65 87 1 3 85 35 55 52 76 94 The invariant factors of this 10 by 10 matrix are 1, 1, 1, 1, 1, 1, 1, 1, 1, 1282266779938614837. ⎛ ⎞ 527 855279206934551 19 ⎜ ⎟ ⎜ ⎟ ⎜57 40 62 92 41 45 64 6 5 9 33⎟ ⎜ ⎟ ⎜15 90 81 96 77 97 64 30 42 8 92⎟ ⎜ ⎟ ⎜81 95 88 21 6 91 29 8 24 93 35⎟ ⎜ ⎟ ⎜36 32 52 64 74 97 49 41 44 28 0 ⎟ ⎜ ⎟ ⎜ ⎟ ⎜29 75 42 76 98 90 37 1 88 8 63⎟ ⎜ ⎟ ⎜88 44 88 92 44 74 12 26 2 67 78⎟ ⎜ ⎟ ⎜74 30 26 53 15 37 62 7 56 31 88⎟ ⎜ ⎟ ⎝52 61 21 48 90 94 60 78 72 56 81⎠ 9055904 67416333462087 The invariant factors of this 10 by 11 matrix are 1, 1, 1, 1, 1, 1, 1, 1, 1, 2. The phenomena observed above are explained by the fact that if s1,s2,...,sn are the invariant factors of a matrix A,thenthe product σk = s1s2 ...sk is the gcd of the determinants of all k by k submatrices of A. (These numbers σk are called the determinantal divisors of A.) E.g. for a 10 by 10 “random” matrix, s1s2 ···s9 is the gcd of the 100 determinants of the 9 by 9 submatrices, and this is “probably” 1. The product s1s2 ···s10 is, up to sign, the determinant of A, which is more-or-less large on the average. Invariant factors of the incidence matrices of some finite projec- tive planes: 28 9 9 26 1 PG2(8) 1 , 2 , 4 , 8 , 72 37 18 35 1 PG2(9) 1 , 3 , 9 , 90 Hall(9) 141, 310, 939, 901 dual Hall(9) 141, 310, 939, 901 Hughes(9) 141, 310, 939, 901 order 10∗ 156, 1054, 1101 28 9 9 28 bordered PG2(8) 1 , 2 , 4 , 8 37 18 37 bordered PG2(9) 1 , 3 , 9 bordered Hall(9)/dual 141, 310, 941 bordered Hughes(9) 141, 310, 941 bordered order 10∗ 156, 1056 6 6 Here is the 2 by 3 inclusion matrix of the 2-subsets versus the 3-subsets of a 6-set. The diagonal entries of one diagonal form are 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 3 ⎛ ⎞ ⎜ 11001000001000000000⎟ ⎜ 10100100000100000000⎟ ⎜ 10010010000010000000⎟ ⎜ ⎟ ⎜ 01100001000001000000⎟ ⎜ 01010000100000100000⎟ ⎜ 00110000010000010000⎟ ⎜ 00001101000000001000⎟ ⎜ ⎟ ⎜ 00001010100000000100⎟ ⎜ 00000110010000000010⎟ ⎜ 00000001110000000001⎟ ⎜ ⎟ ⎜ 00000000001101001000⎟ ⎜ 00000000001010100100⎟ ⎝ 00000000000110010010⎠ 00000000000001110001 00000000000000001111 The invariant factors of the traingular graph T (n) (the line graph L(Kn) of the complete graph) are [see Brouwer and Van Eijl]: − − − − (1)n 2, 2(n 2)(n 3)/2, (2n − 8)n 2, (n − 2)(n − 4) if n ≥ 4 is even, − − − − (1)n 1, 2(n 1)(n 4)/2, (2n − 8)n 2, 2(n − 2)(n − 4) if n ≥ 5 is odd. (1) T (n) is strongly regular and determined up to isomorphism by its parameters except when t = 8, in which case there are three other SRGs (called the Chang graphs) with the same parameters. The invariant factors of the Chang graphs are 18, 212, 87, 241. In these notes, module will always mean Z-module, i.e. a module over the ring Z of integers. These may also be called lattices. Let A be an r by m integer matrix. We use rowZ(A) to denote the module generated by the rows of A, a submodule of Zm; similarly, colZ(A) will denote the module generated by the columns of A, a submodule of Zr. Suppose D = EAF is a diagonal form for A,whereE and F are unimodular. Then A has the same row-module as DF −1;that is, a Z-spanning set for rowZ(A) consists of the vectors d1f1,d2f2, ..., dmfm (2) −1 where fi is the i-th row of F . The vectors f1,...,fm form a m Z-basis for Z ;thefi’s for which di =0forma Z-basis for the integer vectors in the row space of A.AZ-basis for rowZ(A) consists of those vectors difi where di =0. Proposition 1 If v is an integer vector and g is the lcm of the nonzero di’s, then gv ∈ rowZ(A).Ifv is an integer vector in the row space of A,andg is the lcm of the nonzero di’s, then gv ∈ rowZ(A). Example: ⎛ ⎞ 0 −13 10 314⎜ ⎟ 100 ⎝1 −1 −1⎠ = . 21 4 −27 050 01−2 −1 A Z-basis for rowZ(A) consists of the first two rows of DF , and these are (3, 1, 4) and 5(2, 0, 3). The p-rank of A (the rank of A over the field Fp) is 2 except that the 5-rank is only 1. The map a1f1 + ···+ amfm → (a1(mod d1),...,am(mod dm)) is a homomorphism with kernel rowZ(A), so ∼ Zm row ( ) Z ⊕ Z ⊕···⊕Z (3) / Z A = d1 d2 dr. Here Z0 = Z. Let s1,s2,...,sn be the invariant factors of a square integer ma- trix A. Note that if A is nonsingular, then sn is the least value of t so that tA−1 is integral. One way to see this is to use the formula − 1 adj A 1 = A det(A) where Aadj is the classical adjoint of A,with(i, j)-entry i+j (−1) det(Aji), and where Aji is the result of deleting row j and column i from A. The determinant det(Aji) is an integer divisible by s1s2 ···sn−1 and det(A)=s1 ···sn. Another way to understand is to use the fact that that sn is the lcm of the invariant factors and Proposition 1. The relation AB = I means that each column of I is a rational linear com- bination of the columns of A, so that the columns of snI are integer linear combinations of the columns of A. 2. Solutions of linear equations in integers Diagonal forms are related to solutions of systems of linear equa- tions or congruences in integers. This, in fact, was the topic of H. J. S. Smith’s original paper on the subject. Let A be an r by m integer matrix. Suppose EAF = D where E and F are unimodular and D is diagonal with diagonal entries −1 d1,d2,.... The system Ax = b is equivalent to (AF )(F x)=b, and this has integer solutions x if and only if (AF )z = b has an integer solution z. This in turn will have integer solution if and only if EAFz = Eb,orDz = Eb, has integer solutions. In other words, if we let ei denote the i-th row of E, the system Ax = b has integer solutions if and only if eib ≡ 0(moddi)fori =1, 2,...,r. (4) As a simple example, ⎛ ⎞ 0 −13 10 314⎜ ⎟ 100 ⎝1 −1 −1⎠ = 21 4 −27 050 01−2 and so the system of equations 3x + y +4z = a 4x − 2y +7z = b has an integer solution if and only if a ≡ 0(mod1)(thatis,a is an integer) and 2a + b ≡ 0(mod5).
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