Best Known (141, 141+33, s)-Nets in Base 4
(141, 141+33, 1126)-Net over F4 — Constructive and digital
Digital (141, 174, 1126)-net over F4, using
- (u, u+v)-construction [i] based on
- digital (26, 42, 98)-net over F4, using
- trace code for nets [i] based on digital (5, 21, 49)-net over F16, using
- net from sequence [i] based on digital (5, 48)-sequence over F16, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F16 with g(F) = 5 and N(F) ≥ 49, using
- net from sequence [i] based on digital (5, 48)-sequence over F16, using
- trace code for nets [i] based on digital (5, 21, 49)-net over F16, using
- digital (99, 132, 1028)-net over F4, using
- trace code for nets [i] based on digital (0, 33, 257)-net over F256, using
- net from sequence [i] based on digital (0, 256)-sequence over F256, using
- generalized Faure sequence [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F256 with g(F) = 0 and N(F) ≥ 257, using
- the rational function field F256(x) [i]
- Niederreiter sequence [i]
- net from sequence [i] based on digital (0, 256)-sequence over F256, using
- trace code for nets [i] based on digital (0, 33, 257)-net over F256, using
- digital (26, 42, 98)-net over F4, using
(141, 141+33, 9455)-Net over F4 — Digital
Digital (141, 174, 9455)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4174, 9455, F4, 33) (dual of [9455, 9281, 34]-code), using
- discarding factors / shortening the dual code based on linear OA(4174, 16404, F4, 33) (dual of [16404, 16230, 34]-code), using
- construction X applied to C([0,16]) ⊂ C([0,14]) [i] based on
- linear OA(4169, 16385, F4, 33) (dual of [16385, 16216, 34]-code), using the expurgated narrow-sense BCH-code C(I) with length 16385 | 414−1, defining interval I = [0,16], and minimum distance d ≥ |{−16,−15,…,16}|+1 = 34 (BCH-bound) [i]
- linear OA(4155, 16385, F4, 29) (dual of [16385, 16230, 30]-code), using the expurgated narrow-sense BCH-code C(I) with length 16385 | 414−1, defining interval I = [0,14], and minimum distance d ≥ |{−14,−13,…,14}|+1 = 30 (BCH-bound) [i]
- linear OA(45, 19, F4, 3) (dual of [19, 14, 4]-code or 19-cap in PG(4,4)), using
- discarding factors / shortening the dual code based on linear OA(45, 41, F4, 3) (dual of [41, 36, 4]-code or 41-cap in PG(4,4)), using
- construction X applied to C([0,16]) ⊂ C([0,14]) [i] based on
- discarding factors / shortening the dual code based on linear OA(4174, 16404, F4, 33) (dual of [16404, 16230, 34]-code), using
(141, 141+33, 7331458)-Net in Base 4 — Upper bound on s
There is no (141, 174, 7331459)-net in base 4, because
- 1 times m-reduction [i] would yield (141, 173, 7331459)-net in base 4, but
- the generalized Rao bound for nets shows that 4m ≥ 143 343897 322655 586413 967561 815026 256061 947927 788797 015224 033620 947012 072538 562862 386329 100877 966289 881304 > 4173 [i]