Best Known (178−19, 178, s)-Nets in Base 4
(178−19, 178, 932071)-Net over F4 — Constructive and digital
Digital (159, 178, 932071)-net over F4, using
- (u, u+v)-construction [i] based on
- digital (0, 9, 5)-net over F4, using
- net from sequence [i] based on digital (0, 4)-sequence over F4, using
- generalized Faure sequence [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F4 with g(F) = 0 and N(F) ≥ 5, using
- the rational function field F4(x) [i]
- Niederreiter sequence [i]
- net from sequence [i] based on digital (0, 4)-sequence over F4, using
- digital (150, 169, 932066)-net over F4, using
- net defined by OOA [i] based on linear OOA(4169, 932066, F4, 19, 19) (dual of [(932066, 19), 17709085, 20]-NRT-code), using
- OOA 9-folding and stacking with additional row [i] based on linear OA(4169, 8388595, F4, 19) (dual of [8388595, 8388426, 20]-code), using
- discarding factors / shortening the dual code based on linear OA(4169, large, F4, 19) (dual of [large, large−169, 20]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 16777217 | 424−1, defining interval I = [0,9], and minimum distance d ≥ |{−9,−8,…,9}|+1 = 20 (BCH-bound) [i]
- discarding factors / shortening the dual code based on linear OA(4169, large, F4, 19) (dual of [large, large−169, 20]-code), using
- OOA 9-folding and stacking with additional row [i] based on linear OA(4169, 8388595, F4, 19) (dual of [8388595, 8388426, 20]-code), using
- net defined by OOA [i] based on linear OOA(4169, 932066, F4, 19, 19) (dual of [(932066, 19), 17709085, 20]-NRT-code), using
- digital (0, 9, 5)-net over F4, using
(178−19, 178, 4439455)-Net over F4 — Digital
Digital (159, 178, 4439455)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4178, 4439455, F4, 19) (dual of [4439455, 4439277, 20]-code), using
- discarding factors / shortening the dual code based on linear OA(4178, large, F4, 19) (dual of [large, large−178, 20]-code), using
- 9 times code embedding in larger space [i] based on linear OA(4169, large, F4, 19) (dual of [large, large−169, 20]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 16777217 | 424−1, defining interval I = [0,9], and minimum distance d ≥ |{−9,−8,…,9}|+1 = 20 (BCH-bound) [i]
- 9 times code embedding in larger space [i] based on linear OA(4169, large, F4, 19) (dual of [large, large−169, 20]-code), using
- discarding factors / shortening the dual code based on linear OA(4178, large, F4, 19) (dual of [large, large−178, 20]-code), using
(178−19, 178, large)-Net in Base 4 — Upper bound on s
There is no (159, 178, large)-net in base 4, because
- 17 times m-reduction [i] would yield (159, 161, large)-net in base 4, but