A New Lower Bound for A(17,6,6)

A New Low er Bound for A (17 , 6 , 6) Y e ow Meng Che e In t eractiv e Dig it a l Media Program Office Media Dev elopment Authorit y 140 Hill Street Singap ore 1793 69 and Division of Mathematical Science s Sc ho ol of Ph ysical a nd Mathematical Sciences Nan y ang T ec hnological Univ ersity Singap ore 6376 16 Abstract W e construct a r ecord-br e aking binary co de o f length 17, minimal distance 6, constant w eight 6, and containing 113 co dewords. 1 In tro duction Let A ( n, d, w ) denote the max imum p o ssible num b er of co dewords in a binary co de o f length n , minimal dis ta nce d and consta nt weigh t w . The Nor dstrom-Robinso n co de N 16 of length 16, minimal distance 6, and containing 256 co dewords has w eight enumerator 1 + 112 x 6 + 30 x 8 + 112 x 10 + x 16 . Hence, taking all the co dewords o f weigh t 6 in N 16 gives a constant weight co de that shows A (16 , 6 , 6 ) ≥ 112 . Since A (17 , 6 , 6) ≥ A (16 , 6 , 6), we also hav e A (17 , 6 , 6) ≥ 112. This is in fact the b e st low er b ound o n A (17 , 6 , 6) known [2]. In this note, w e give the fir st improvemen t on the low e r b ound for A (17 , 6 , 6) since that implied by the 196 7 r esult of Nordstrom and Robinson [3]. W e ex hibit a new binary co de C of length 1 7, minimal distance 6, constant weigh t 6, and containing 113 co dewords, showing A (17 , 6 , 6) ≥ 113. O ur co de ha s no particula r structure (its automor - phism group is triv ia l) and is obtained through a combination of search techn iques inv olving simulated annealing [4], length-r eduction [1], and lo cal optimization. 1 The s upp or t supp( x ) of a co deword x = ( x 1 , . . . , x n ) is the set of indices of its non-zer o co o rdinates, that is, supp( x ) = { i | x i 6 = 0 } . The suppor ts of the co dewords in C ar e listed in the next s ection. 2 The Co de 0 1 2 3 6 15 0 1 2 4 11 16 0 1 2 7 8 9 0 1 2 10 12 13 0 1 3 4 8 10 0 1 3 5 7 12 0 1 3 9 13 16 0 1 4 6 7 13 0 1 5 6 10 16 0 1 5 8 11 13 0 1 6 9 11 12 0 1 7 10 11 15 0 1 8 12 14 15 0 2 3 4 9 12 0 2 3 5 8 16 0 2 3 7 11 13 0 2 4 5 7 10 0 2 4 8 13 15 0 2 5 6 9 13 0 2 5 11 14 15 0 2 6 7 12 16 0 2 6 8 10 11 0 3 4 5 6 11 0 3 4 7 14 16 0 3 5 10 13 15 0 3 6 7 9 10 0 3 6 8 12 13 0 3 8 9 11 15 0 3 10 11 12 14 0 4 5 12 13 14 0 4 6 8 9 16 0 4 6 10 12 15 0 4 7 8 11 12 0 4 9 10 11 13 0 5 6 7 8 15 0 5 8 9 10 14 0 5 9 12 15 16 0 6 11 13 14 16 0 7 8 10 13 16 0 7 9 13 14 15 1 2 3 4 5 13 1 2 3 7 10 14 1 2 3 8 11 12 1 2 4 6 9 10 1 2 4 7 12 15 1 2 5 6 7 11 1 2 5 8 10 15 1 2 5 12 14 16 1 2 6 8 13 16 1 2 9 11 13 15 1 3 4 6 12 16 1 3 4 7 9 11 1 3 5 6 8 14 1 3 5 11 15 16 1 3 6 10 11 13 1 3 7 8 13 15 1 3 9 10 12 15 1 4 5 7 8 16 1 4 5 9 14 15 1 4 5 10 11 12 1 4 6 8 11 15 1 4 8 9 12 13 1 4 10 13 14 16 1 5 6 12 13 15 1 5 7 9 10 13 1 6 7 8 10 12 1 6 7 9 15 16 1 7 11 12 13 16 1 8 9 10 11 16 2 3 4 6 7 8 2 3 4 10 11 15 2 3 5 6 10 12 2 3 5 7 9 15 2 3 6 9 11 16 2 3 8 9 10 13 2 3 12 13 15 16 2 4 5 6 15 16 2 4 5 8 9 11 2 4 6 11 12 13 2 4 7 9 13 16 2 4 8 10 12 14 2 5 7 8 12 13 2 5 10 11 13 16 2 6 7 10 13 15 2 6 8 9 12 15 2 7 8 11 15 16 2 7 9 10 11 12 2 9 10 14 15 16 3 4 5 8 12 15 3 4 5 9 10 16 3 4 6 13 14 15 3 4 7 10 12 13 3 4 8 11 13 16 3 5 6 7 13 16 3 5 7 8 10 11 3 5 9 11 12 13 3 6 7 11 12 15 3 6 8 10 15 16 3 7 8 9 12 16 4 5 6 7 9 12 4 5 6 8 10 13 4 5 7 11 13 15 4 6 7 10 11 14 4 7 8 9 10 15 4 11 12 14 15 16 5 6 8 11 12 16 5 6 9 10 11 15 5 7 9 11 14 16 5 7 10 12 14 15 5 8 13 14 15 16 6 7 8 9 11 13 6 9 10 12 13 16 8 10 11 12 13 15 2 References [1] H. K. Aw, Y. M. Chee and A. C. H. Ling, ”Six new constan t w eigh t binary co des”, Ars Combin. , 6 7 (2003), 313–31 8. [2] A. E. Brouw er , J. B. Shearer , N. J. A. Sloane and W. D. Smith, ”A new table o f cons ta nt weigh t c o des”, IEE E T ra ns. Inform. The ory , 36 (199 0 ), 13 34–13 80. [3] A. W. Nor dstrom and J. P . Robinso n, ”An o ptim um nonlinea r co de”, Inform. Contr ol , 11 (19 6 7), 613–6 16. [4] K. J . Nurmela and P . R. J. ¨ Osterg ˚ ard, Upp er b oun ds for c overing designs by simulate d anne aling , Congr . Numer. bf 9 6 (1993 ), 93– 111. 3