- Email: [email protected]
A short proof of a conjecture on quasi-symmetric 3-designs
Discrete Mathematics 96 (1991) 71-74 North-Holland
71
Note
A short proof of a conjecture on quasi-symmetric 3-designs Rajendra M. Pawale and Sharad S. Sane Departmentof Mathematics, Universityof Bombay, Viayanagari, Bombay-400098, India Received 21 July 1989 Revised 15 December 1989
Abstract Pawale, R.M. and S.S. Sane, A short p,oof of a conjecture on quasi-symmetric 2-designs, Discrete Mathematics 96 (1991) 71-74. It was conjettured by Sane and M.S. Shrikhande that the only nontrivial quasi-symmetric 3-design with the smaller block intersection number one is either the Witt 4-(23, 7, 1) design or its residual. Calderbank and Morton recently proved this conjecture using sophisticated number theoretic arguments. A short and elementary proof of this conjecture is presented in this paper.
1. Introduction
The purpose of this paper is to give a simple, elementary and short proof of the following result which was recently proved by Calderbank and Morton. Theorem [4, conjecture 4.5 and 2, theorem 1). Let D be a quasi-symmetric 3-design with the smaller block intersection n u m b e r x = 1. Then D is either the V¢itt 4-(23, 7, 1) design or its residual (treating the 3-(5, 3, 1) design as trivial).
After preparing an earlier version, the authors learnt that Calderbank and Morton [2] recently proved the conjecture. Their proof, though similar in theme, uses sophisticated arguments in number theory. In c,mtrast, our proof is both shorter and very elementary. As in our case, Calderbank and Morton also obtain = y ~ 5 or y ~<4 (where y is the other intersection numbe 0. However, to dispose these two cases off, they find all the integer points on two elliptic curves which involves long and sophisticated arguments. Our proof is based on the 0012-365X/91/$03.50 © 1991 --Elsevier Science Publishers B.V. All rights reserved
R.M. Pawale, S.S. Sane
72
quadratic equation f().) = A;~ 2 + BZ + C = 0 ( E q u a t i o n (7)) w h e r e A , B and C are polynomials in k and y. The p r o o f is essentially b r o k e n in three parts: y = 3 (which has been dealt in [4, 2]), y = 4 and y I> 5 where we force Z = y.
2. The proof
O u r starting point is the following fundamental identity valid for all quasisymmetric 3-designs with the smaller block intersection n u m b e r x t> 1: Z[(k - 1)2(0 - 2){k 2 - 2te(v - 1)} + a~(a~ - 1)(v - 1)(v - 2) 2] = k ( k - 1)(k - 2)(k - tr)[(k - 1) z - tr(v - 2)1
where tr = x o r y .
(1)
Neumaier [3] seems to have first o b t a i n e d an equation which is equivalent to (1); a p r o o f of (1) m a y be f o u n d in [4]. F r o m this p o i n t o n w e s p e c i a l i z e t o x = 1. Then as pointed out in [4], (1) obtains (after the substitution tr = 1): x(v - 2) k(k-2)=
(k - 1) 2 - (v - 2) k 2-2(0-1)
(2)
For the sake of completeness, we dispose of y ~< 3, which aas b e e n dealt at length in [4, 2]. If y = 2, then ,a = 1 and (2) becomes: 2(v - 2) 2 - 2[k(k - 1) - 1](v - 2) + k ( k - 1)2(k - 2) = 0
(3)
which has a non-negative solution only if k = 3 and v = 5. This is the trivial 3-(5, 3, 1) design. N o w let y / > 3. T h e n a simple two-way counting p r o d u c e s (k - 2)(~, - 1) = (~.2 - 1)(y - 2).
(4)
We note an immediate consequence o f (4). In a 2-design D ' o b t a i n e d as a contraction of D, we have k ' < r ' (since x 4: 0), i.e., k - 1 < ~,2- T h e r e f o r e (4) implies: X~>y.
(5)
N o w invoke the fundamental identity ;~2 = ~.(v - 2 ) / ( k - 2) and use it in (4) to obtain: v - 2 = ( k - 2)2~, - ( k - 2 ) ( k - y )
).(y - 2)
(6)
Substitution o f this value of v - 2 in (2) results in the equation: f(;t) = AX 2 + B2 + C = 0.
(7)
where A = k 2 ( y - 4) + 8k - 2(y + 2),
B = -y[kZ(y
- 4) - k ( y - I0) - ( y + 6)]
On quasi-s)mmetric 3-designs
73
and C = - y [ k 2 - k ( y + 2) + 2el.
If y = 3 then the condition y - 1 divides k - 1 implie~ that k is odd and (7) becomes: (k 2 - 8k + 10)$ 2 - 3(k 2 - 7k + 9)$ + 3(k - 2)(k - 3) = 0
(8)
whose discriminant is negative for k I> 9. Also no integer solution is possible for k = 3 or 5. So k = 7 and $--= 5 or 4 which by (6) gives v = 23 or 22 respectively. Since y = 3, counting the occurrences of 4-tuples finishes the proof in the first case; $ = 4 is also similar and has been completely setded in [4] where it is shown that (x, y) = (1, 3) if and only if D is the Witt 4-(23, 7, l) design or its residual. This is also essentially clear using the well-known fact that Witt designs are uniquely determined by their parameters. Let therefore, y equal 4. Then (7) reduces to a mopic equation g ( k ) = k 2 - 2($ 2 - 35 + 3)k + ( 3 , ~ 2 - 103. + 8) = 0 whose discriminant is 4A 1 where A1, in view of (5) lies strictly between the two perfect squares ( $ 2 - 3 5 + 1) 2 and ($2 _ 35 + 2) 2. So this equation has no integral solution for k. Finally assume that y >i 5, look at (7) again and observe that A is positive while B and C are both negative. Hence we must take the larger (positive) root in (7). If - B > y A then - ( y - 2 ) ( k - 1) is positive which is false. Similarly, if - C > y A then 4k + 4 >1 4(y + 1) > k 2 ( y - 5) + k ( y + 10) ~> 15k, which indeed is false. So we have y >! - B / A and y >~ - C / A . Hence the positive solution of (7) for $ is: $ = ( - B I A + V~2)/2
where A 2 = ( - B / A ) z + 4 ( - C / A )
Hence 3. ~< (y + V ~ + 4y)/2 < y + 1. Use of (5) obtains ~, = y. Substitution then of )~ = y in (7) obtains a monic quadratic h ( k ) = k 2 - (y2 _ y + 2)k + y 2 = 0.
(9)
The final blow is equally easy: the discriminant of (9) lies strictly between (using y >/5) the two perfect squares (y2 _ y _ 1)2 and (y2 _ y)2 and hence the monic (9) has no integral solution.
Acknowledgement The authors should like to thank Dr. A.R. Calderbank for his suggestions on the earlier version of the paper.
74
R,M. Pawale, S.S. Sane
References 1~] A.R. Calderbank, Inequalities for quasi-symmetric designs, J. Combin. Theory Ser. A 48 (1988) 53-63. [2j A.IL C~ddcrbank and P. Morton, Quasi-symmetric 3-designs and elliptic curves, SIAM J. Discrete Math., to appear. 13] A. Neumaier, Regular sets and quasi-symmetric 2-designs, in: D. Jungnickel and K. Vedder, eds., Combinatorial Theory, Lecture Notes in Math. No. 969 (Springer, Berlin, 1982) 258-275. [4] S.S. Sane and M.S. Shrikhande, Quasi-symmetric 2, 3 and 4-designs, Combinatorica 7 (1987) 291-301. [5] M.S. Shrikhande, Designs intersection numbers and codes (preprint), IMA Volume on Mathematics and its Applications, Vol. 21, Coding Theory and Designs, Part II: Design Theory (1988) to appear.
71
Note
A short proof of a conjecture on quasi-symmetric 3-designs Rajendra M. Pawale and Sharad S. Sane Departmentof Mathematics, Universityof Bombay, Viayanagari, Bombay-400098, India Received 21 July 1989 Revised 15 December 1989
Abstract Pawale, R.M. and S.S. Sane, A short p,oof of a conjecture on quasi-symmetric 2-designs, Discrete Mathematics 96 (1991) 71-74. It was conjettured by Sane and M.S. Shrikhande that the only nontrivial quasi-symmetric 3-design with the smaller block intersection number one is either the Witt 4-(23, 7, 1) design or its residual. Calderbank and Morton recently proved this conjecture using sophisticated number theoretic arguments. A short and elementary proof of this conjecture is presented in this paper.
1. Introduction
The purpose of this paper is to give a simple, elementary and short proof of the following result which was recently proved by Calderbank and Morton. Theorem [4, conjecture 4.5 and 2, theorem 1). Let D be a quasi-symmetric 3-design with the smaller block intersection n u m b e r x = 1. Then D is either the V¢itt 4-(23, 7, 1) design or its residual (treating the 3-(5, 3, 1) design as trivial).
After preparing an earlier version, the authors learnt that Calderbank and Morton [2] recently proved the conjecture. Their proof, though similar in theme, uses sophisticated arguments in number theory. In c,mtrast, our proof is both shorter and very elementary. As in our case, Calderbank and Morton also obtain = y ~ 5 or y ~<4 (where y is the other intersection numbe 0. However, to dispose these two cases off, they find all the integer points on two elliptic curves which involves long and sophisticated arguments. Our proof is based on the 0012-365X/91/$03.50 © 1991 --Elsevier Science Publishers B.V. All rights reserved
R.M. Pawale, S.S. Sane
72
quadratic equation f().) = A;~ 2 + BZ + C = 0 ( E q u a t i o n (7)) w h e r e A , B and C are polynomials in k and y. The p r o o f is essentially b r o k e n in three parts: y = 3 (which has been dealt in [4, 2]), y = 4 and y I> 5 where we force Z = y.
2. The proof
O u r starting point is the following fundamental identity valid for all quasisymmetric 3-designs with the smaller block intersection n u m b e r x t> 1: Z[(k - 1)2(0 - 2){k 2 - 2te(v - 1)} + a~(a~ - 1)(v - 1)(v - 2) 2] = k ( k - 1)(k - 2)(k - tr)[(k - 1) z - tr(v - 2)1
where tr = x o r y .
(1)
Neumaier [3] seems to have first o b t a i n e d an equation which is equivalent to (1); a p r o o f of (1) m a y be f o u n d in [4]. F r o m this p o i n t o n w e s p e c i a l i z e t o x = 1. Then as pointed out in [4], (1) obtains (after the substitution tr = 1): x(v - 2) k(k-2)=
(k - 1) 2 - (v - 2) k 2-2(0-1)
(2)
For the sake of completeness, we dispose of y ~< 3, which aas b e e n dealt at length in [4, 2]. If y = 2, then ,a = 1 and (2) becomes: 2(v - 2) 2 - 2[k(k - 1) - 1](v - 2) + k ( k - 1)2(k - 2) = 0
(3)
which has a non-negative solution only if k = 3 and v = 5. This is the trivial 3-(5, 3, 1) design. N o w let y / > 3. T h e n a simple two-way counting p r o d u c e s (k - 2)(~, - 1) = (~.2 - 1)(y - 2).
(4)
We note an immediate consequence o f (4). In a 2-design D ' o b t a i n e d as a contraction of D, we have k ' < r ' (since x 4: 0), i.e., k - 1 < ~,2- T h e r e f o r e (4) implies: X~>y.
(5)
N o w invoke the fundamental identity ;~2 = ~.(v - 2 ) / ( k - 2) and use it in (4) to obtain: v - 2 = ( k - 2)2~, - ( k - 2 ) ( k - y )
).(y - 2)
(6)
Substitution o f this value of v - 2 in (2) results in the equation: f(;t) = AX 2 + B2 + C = 0.
(7)
where A = k 2 ( y - 4) + 8k - 2(y + 2),
B = -y[kZ(y
- 4) - k ( y - I0) - ( y + 6)]
On quasi-s)mmetric 3-designs
73
and C = - y [ k 2 - k ( y + 2) + 2el.
If y = 3 then the condition y - 1 divides k - 1 implie~ that k is odd and (7) becomes: (k 2 - 8k + 10)$ 2 - 3(k 2 - 7k + 9)$ + 3(k - 2)(k - 3) = 0
(8)
whose discriminant is negative for k I> 9. Also no integer solution is possible for k = 3 or 5. So k = 7 and $--= 5 or 4 which by (6) gives v = 23 or 22 respectively. Since y = 3, counting the occurrences of 4-tuples finishes the proof in the first case; $ = 4 is also similar and has been completely setded in [4] where it is shown that (x, y) = (1, 3) if and only if D is the Witt 4-(23, 7, l) design or its residual. This is also essentially clear using the well-known fact that Witt designs are uniquely determined by their parameters. Let therefore, y equal 4. Then (7) reduces to a mopic equation g ( k ) = k 2 - 2($ 2 - 35 + 3)k + ( 3 , ~ 2 - 103. + 8) = 0 whose discriminant is 4A 1 where A1, in view of (5) lies strictly between the two perfect squares ( $ 2 - 3 5 + 1) 2 and ($2 _ 35 + 2) 2. So this equation has no integral solution for k. Finally assume that y >i 5, look at (7) again and observe that A is positive while B and C are both negative. Hence we must take the larger (positive) root in (7). If - B > y A then - ( y - 2 ) ( k - 1) is positive which is false. Similarly, if - C > y A then 4k + 4 >1 4(y + 1) > k 2 ( y - 5) + k ( y + 10) ~> 15k, which indeed is false. So we have y >! - B / A and y >~ - C / A . Hence the positive solution of (7) for $ is: $ = ( - B I A + V~2)/2
where A 2 = ( - B / A ) z + 4 ( - C / A )
Hence 3. ~< (y + V ~ + 4y)/2 < y + 1. Use of (5) obtains ~, = y. Substitution then of )~ = y in (7) obtains a monic quadratic h ( k ) = k 2 - (y2 _ y + 2)k + y 2 = 0.
(9)
The final blow is equally easy: the discriminant of (9) lies strictly between (using y >/5) the two perfect squares (y2 _ y _ 1)2 and (y2 _ y)2 and hence the monic (9) has no integral solution.
Acknowledgement The authors should like to thank Dr. A.R. Calderbank for his suggestions on the earlier version of the paper.
74
R,M. Pawale, S.S. Sane
References 1~] A.R. Calderbank, Inequalities for quasi-symmetric designs, J. Combin. Theory Ser. A 48 (1988) 53-63. [2j A.IL C~ddcrbank and P. Morton, Quasi-symmetric 3-designs and elliptic curves, SIAM J. Discrete Math., to appear. 13] A. Neumaier, Regular sets and quasi-symmetric 2-designs, in: D. Jungnickel and K. Vedder, eds., Combinatorial Theory, Lecture Notes in Math. No. 969 (Springer, Berlin, 1982) 258-275. [4] S.S. Sane and M.S. Shrikhande, Quasi-symmetric 2, 3 and 4-designs, Combinatorica 7 (1987) 291-301. [5] M.S. Shrikhande, Designs intersection numbers and codes (preprint), IMA Volume on Mathematics and its Applications, Vol. 21, Coding Theory and Designs, Part II: Design Theory (1988) to appear.