Best Known (211−45, 211, s)-Nets in Base 4
(211−45, 211, 1056)-Net over F4 — Constructive and digital
Digital (166, 211, 1056)-net over F4, using
- 1 times m-reduction [i] based on digital (166, 212, 1056)-net over F4, using
- trace code for nets [i] based on digital (7, 53, 264)-net over F256, using
- net from sequence [i] based on digital (7, 263)-sequence over F256, using
- trace code for nets [i] based on digital (7, 53, 264)-net over F256, using
(211−45, 211, 4483)-Net over F4 — Digital
Digital (166, 211, 4483)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4211, 4483, F4, 45) (dual of [4483, 4272, 46]-code), using
- 375 step Varšamov–Edel lengthening with (ri) = (3, 1, 0, 0, 1, 5 times 0, 1, 9 times 0, 1, 17 times 0, 1, 29 times 0, 1, 45 times 0, 1, 64 times 0, 1, 86 times 0, 1, 108 times 0) [i] based on linear OA(4199, 4096, F4, 45) (dual of [4096, 3897, 46]-code), using
- an extension Ce(44) of the primitive narrow-sense BCH-code C(I) with length 4095 = 46−1, defining interval I = [1,44], and designed minimum distance d ≥ |I|+1 = 45 [i]
- 375 step Varšamov–Edel lengthening with (ri) = (3, 1, 0, 0, 1, 5 times 0, 1, 9 times 0, 1, 17 times 0, 1, 29 times 0, 1, 45 times 0, 1, 64 times 0, 1, 86 times 0, 1, 108 times 0) [i] based on linear OA(4199, 4096, F4, 45) (dual of [4096, 3897, 46]-code), using
(211−45, 211, 1685246)-Net in Base 4 — Upper bound on s
There is no (166, 211, 1685247)-net in base 4, because
- 1 times m-reduction [i] would yield (166, 210, 1685247)-net in base 4, but
- the generalized Rao bound for nets shows that 4m ≥ 2 707699 692436 455692 302489 759339 697537 000222 881237 379216 642773 753828 971958 733229 680186 583958 140843 663704 362068 405428 649366 512335 > 4210 [i]