Codes and modules associated with designs and t -uniform hypergraphs - PowerPoint PPT Presentation

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 N t (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 , of orders r and m , so that EAF = 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 − 1 3 � � � � � 1 0 3 1 4 1 0 0 1 − 1 − 1 ⎠ = . ⎜ ⎟ 2 1 4 − 2 7 0 5 0 ⎝ 0 1 − 2 The matrix on the right is a diagonal form for the middle matrix on the left.

Let the diagonal entries of D be d 1 , d 2 , d 3 , . . . . ⎛ · · · ⎞ 2 0 0 · · · 0 24 0 ⎜ ⎟ ⎜ ⎟ 120 · · · 0 0 ⎜ ⎟ ⎝ ⎠ . . . ... . . . . . . If all diagonal entries d i are nonnegative and d i divides d i +1 for i = 1 , 2 , . . . , then D is called the integer Smith normal form of A , or simply the Smith form of A , and the integers d i are called the invariant factors , or the elementary divisors of A . The Smith 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, d i 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.

� � � � � � 3 1 4 3 1 4 3 1 1 → → − 2 − 3 − 3 4 7 1 3 1 2 � � � � � � 3 0 1 0 0 1 0 0 1 → → → 1 − 5 2 − 5 5 2 0 5 2 � � � � 0 0 1 1 0 0 → → 0 5 0 0 5 0

⎛ ⎞ 2 85 36 24 96 60 54 70 95 80 50 46 82 88 94 25 2 1 21 40 ⎜ ⎟ ⎜ ⎟ 85 86 52 4 24 45 57 94 38 58 ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ 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 .

⎛ ⎞ 52 7 85 52 79 20 69 34 55 1 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 ⎜ ⎟ ⎝ ⎠ 90 55 90 4 67 41 63 33 46 20 87 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 s 1 , s 2 , . . . , s n are the invariant factors of a matrix A , then the product σ k = s 1 s 2 . . . s k is the gcd of the determinants of all k by k submatrices of A . (These numbers σ k are called the determinantal divisors of A .) E.g. for a 10 by 10 “random” matrix, s 1 s 2 · · · s 9 is the gcd of the 100 determinants of the 9 by 9 submatrices, and this is “probably” 1. The product s 1 s 2 · · · s 10 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: 1 28 , 2 9 , 4 9 , 8 26 , 72 1 PG 2 (8) 1 37 , 3 18 , 9 35 , 90 1 PG 2 (9) 1 41 , 3 10 , 9 39 , 90 1 Hall(9) 1 41 , 3 10 , 9 39 , 90 1 dual Hall(9) 1 41 , 3 10 , 9 39 , 90 1 Hughes(9) order 10 ∗ 1 56 , 10 54 , 110 1 1 28 , 2 9 , 4 9 , 8 28 bordered PG 2 (8) 1 37 , 3 18 , 9 37 bordered PG 2 (9) 1 41 , 3 10 , 9 41 bordered Hall(9)/dual 1 41 , 3 10 , 9 41 bordered Hughes(9) bordered order 10 ∗ 1 56 , 10 56

� 6 � 6 � � Here is the by inclusion matrix of the 2-subsets versus 2 3 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 ⎛ ⎞ 1 1 0 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 1 0 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 ⎜ ⎟ 1 0 0 1 0 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 ⎜ ⎟ 0 1 1 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 0 ⎜ ⎟ ⎜ ⎟ 0 1 0 1 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 ⎜ ⎟ 0 0 1 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 ⎜ ⎟ 0 0 0 0 1 1 0 1 0 0 0 0 0 0 0 0 1 0 0 0 ⎜ ⎟ 0 0 0 0 1 0 1 0 1 0 0 0 0 0 0 0 0 1 0 0 ⎜ ⎟ ⎜ ⎟ 0 0 0 0 0 1 1 0 0 1 0 0 0 0 0 0 0 0 1 0 ⎜ ⎟ 0 0 0 0 0 0 0 1 1 1 0 0 0 0 0 0 0 0 0 1 ⎜ ⎟ 0 0 0 0 0 0 0 0 0 0 1 1 0 1 0 0 1 0 0 0 ⎜ ⎟ ⎜ 0 0 0 0 0 0 0 0 0 0 1 0 1 0 1 0 0 1 0 0 ⎟ ⎜ ⎟ 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 1 0 0 1 0 ⎜ ⎟ 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 0 0 1 ⎝ ⎠ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1

The invariant factors of the traingular graph T ( n ) (the line graph L ( K n ) of the complete graph) are [see Brouwer and Van Eijl]: (1) n − 2 , 2 ( n − 2)( n − 3) / 2 , (2 n − 8) n − 2 , ( n − 2)( n − 4) if n ≥ 4 is even, (1) n − 1 , 2 ( n − 1)( n − 4) / 2 , (2 n − 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 1 8 , 2 12 , 8 7 , 24 1 .

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 row Z ( A ) to denote the module generated by the rows of A , a submodule of Z m ; similarly, col Z ( A ) will denote the module generated by the columns of A , a submodule of Z r .

Suppose D = EAF is a diagonal form for A , where E and F are unimodular. Then A has the same row-module as DF − 1 ; that is, a Z -spanning set for row Z ( A ) consists of the vectors d 1 f 1 , d 2 f 2 , . . . , d m f m (2) where f i is the i -th row of F − 1 . The vectors f 1 , . . . , f m form a Z -basis for Z m ; the f i ’s for which d i � = 0 form a Z -basis for the integer vectors in the row space of A . A Z -basis for row Z ( A ) consists of those vectors d i f i where d i � = 0. Proposition 1 If v is an integer vector and g is the lcm of the nonzero d i ’s, then g v ∈ row Z ( A ) . If v is an integer vector in the row space of A , and g ′ is the lcm of the nonzero d i ’s, then g v ∈ row Z ( A ) .

Example: � ⎛ ⎞ 0 − 1 3 � � � � � 1 0 3 1 4 1 0 0 − 1 − 1 1 ⎠ = . ⎜ ⎟ − 2 2 1 4 7 0 5 0 ⎝ − 2 0 1 A Z -basis for row Z ( A ) consists of the first two rows of DF − 1 , and these are (3 , 1 , 4) and 5(2 , 0 , 3). The p -rank of A (the rank of A over the field F p ) is 2 except that the 5-rank is only 1.

The map a 1 f 1 + · · · + a m f m �→ ( a 1 (mod d 1 ) , . . . , a m (mod d m )) is a homomorphism with kernel row Z ( A ), so Z m / row Z ( A ) ∼ = Z d 1 ⊕ Z d 2 ⊕ · · · ⊕ Z d r . (3) Here Z 0 = Z .

Let s 1 , s 2 , . . . , s n be the invariant factors of a square integer ma- trix A . Note that if A is nonsingular, then s n is the least value of t so that tA − 1 is integral. One way to see this is to use the formula 1 A − 1 = det( A ) A adj where A adj is the classical adjoint of A , with ( i, j )-entry ( − 1) i + j det( A ji ), and where A ji is the result of deleting row j and column i from A . The determinant det( A ji ) is an integer divisible by s 1 s 2 · · · s n − 1 and det( A ) = s 1 · · · s n . Another way to understand is to use the fact that that s n 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 s n I are integer linear combinations of the columns of A .