Best Known (170, 170+41, s)-Nets in Base 4
(170, 170+41, 1539)-Net over F4 — Constructive and digital
Digital (170, 211, 1539)-net over F4, using
- 2 times m-reduction [i] based on digital (170, 213, 1539)-net over F4, using
- trace code for nets [i] based on digital (28, 71, 513)-net over F64, using
- net from sequence [i] based on digital (28, 512)-sequence over F64, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F64 with g(F) = 28 and N(F) ≥ 513, using
- the Hermitian function field over F64 [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F64 with g(F) = 28 and N(F) ≥ 513, using
- net from sequence [i] based on digital (28, 512)-sequence over F64, using
- trace code for nets [i] based on digital (28, 71, 513)-net over F64, using
(170, 170+41, 8926)-Net over F4 — Digital
Digital (170, 211, 8926)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4211, 8926, F4, 41) (dual of [8926, 8715, 42]-code), using
- discarding factors / shortening the dual code based on linear OA(4211, 16384, F4, 41) (dual of [16384, 16173, 42]-code), using
- an extension Ce(40) of the primitive narrow-sense BCH-code C(I) with length 16383 = 47−1, defining interval I = [1,40], and designed minimum distance d ≥ |I|+1 = 41 [i]
- discarding factors / shortening the dual code based on linear OA(4211, 16384, F4, 41) (dual of [16384, 16173, 42]-code), using
(170, 170+41, 5805153)-Net in Base 4 — Upper bound on s
There is no (170, 211, 5805154)-net in base 4, because
- 1 times m-reduction [i] would yield (170, 210, 5805154)-net in base 4, but
- the generalized Rao bound for nets shows that 4m ≥ 2 707693 932429 720883 544136 921432 751664 047915 293552 844276 090927 020385 477401 406520 770999 910886 940204 194389 495331 122467 452308 335576 > 4210 [i]