Best Known (162, 162+19, s)-Nets in Base 4
(162, 162+19, 932080)-Net over F4 — Constructive and digital
Digital (162, 181, 932080)-net over F4, using
- (u, u+v)-construction [i] based on
- digital (3, 12, 14)-net over F4, using
- net from sequence [i] based on digital (3, 13)-sequence over F4, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F4 with g(F) = 3 and N(F) ≥ 14, using
- net from sequence [i] based on digital (3, 13)-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 (3, 12, 14)-net over F4, using
(162, 162+19, 5669905)-Net over F4 — Digital
Digital (162, 181, 5669905)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4181, 5669905, F4, 19) (dual of [5669905, 5669724, 20]-code), using
- discarding factors / shortening the dual code based on linear OA(4181, large, F4, 19) (dual of [large, large−181, 20]-code), using
- strength reduction [i] based on linear OA(4181, large, F4, 21) (dual of [large, large−181, 22]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 16777215 = 412−1, defining interval I = [0,20], and designed minimum distance d ≥ |I|+1 = 22 [i]
- strength reduction [i] based on linear OA(4181, large, F4, 21) (dual of [large, large−181, 22]-code), using
- discarding factors / shortening the dual code based on linear OA(4181, large, F4, 19) (dual of [large, large−181, 20]-code), using
(162, 162+19, large)-Net in Base 4 — Upper bound on s
There is no (162, 181, large)-net in base 4, because
- 17 times m-reduction [i] would yield (162, 164, large)-net in base 4, but