Best Known (81, 81+30, s)-Nets in Base 4
(81, 81+30, 531)-Net over F4 — Constructive and digital
Digital (81, 111, 531)-net over F4, using
- trace code for nets [i] based on digital (7, 37, 177)-net over F64, using
- net from sequence [i] based on digital (7, 176)-sequence over F64, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F64 with g(F) = 7 and N(F) ≥ 177, using
- net from sequence [i] based on digital (7, 176)-sequence over F64, using
(81, 81+30, 852)-Net over F4 — Digital
Digital (81, 111, 852)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4111, 852, F4, 30) (dual of [852, 741, 31]-code), using
- discarding factors / shortening the dual code based on linear OA(4111, 1023, F4, 30) (dual of [1023, 912, 31]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 1023 = 45−1, defining interval I = [0,29], and designed minimum distance d ≥ |I|+1 = 31 [i]
- discarding factors / shortening the dual code based on linear OA(4111, 1023, F4, 30) (dual of [1023, 912, 31]-code), using
(81, 81+30, 61066)-Net in Base 4 — Upper bound on s
There is no (81, 111, 61067)-net in base 4, because
- the generalized Rao bound for nets shows that 4m ≥ 6 740589 552823 181026 339361 993229 140045 188211 193902 754972 280497 025792 > 4111 [i]