Best Known (112, 112+33, s)-Nets in Base 4
(112, 112+33, 1040)-Net over F4 — Constructive and digital
Digital (112, 145, 1040)-net over F4, using
- 41 times duplication [i] based on digital (111, 144, 1040)-net over F4, using
- trace code for nets [i] based on digital (3, 36, 260)-net over F256, using
- net from sequence [i] based on digital (3, 259)-sequence over F256, using
- trace code for nets [i] based on digital (3, 36, 260)-net over F256, using
(112, 112+33, 2567)-Net over F4 — Digital
Digital (112, 145, 2567)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4145, 2567, F4, 33) (dual of [2567, 2422, 34]-code), using
- discarding factors / shortening the dual code based on linear OA(4145, 4096, F4, 33) (dual of [4096, 3951, 34]-code), using
- an extension Ce(32) of the primitive narrow-sense BCH-code C(I) with length 4095 = 46−1, defining interval I = [1,32], and designed minimum distance d ≥ |I|+1 = 33 [i]
- discarding factors / shortening the dual code based on linear OA(4145, 4096, F4, 33) (dual of [4096, 3951, 34]-code), using
(112, 112+33, 594221)-Net in Base 4 — Upper bound on s
There is no (112, 145, 594222)-net in base 4, because
- 1 times m-reduction [i] would yield (112, 144, 594222)-net in base 4, but
- the generalized Rao bound for nets shows that 4m ≥ 497 336320 619442 932462 927554 189217 635631 933019 043297 649859 356971 490228 432248 846929 586762 > 4144 [i]