Best Known (27, 27+9, s)-Nets in Base 4
(27, 27+9, 1028)-Net over F4 — Constructive and digital
Digital (27, 36, 1028)-net over F4, using
- trace code for nets [i] based on digital (0, 9, 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
(27, 27+9, 1055)-Net over F4 — Digital
Digital (27, 36, 1055)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(436, 1055, F4, 9) (dual of [1055, 1019, 10]-code), using
- 25 step Varšamov–Edel lengthening with (ri) = (2, 1, 0, 0, 1, 6 times 0, 1, 13 times 0) [i] based on linear OA(431, 1025, F4, 9) (dual of [1025, 994, 10]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 1025 | 410−1, defining interval I = [0,4], and minimum distance d ≥ |{−4,−3,…,4}|+1 = 10 (BCH-bound) [i]
- 25 step Varšamov–Edel lengthening with (ri) = (2, 1, 0, 0, 1, 6 times 0, 1, 13 times 0) [i] based on linear OA(431, 1025, F4, 9) (dual of [1025, 994, 10]-code), using
(27, 27+9, 136756)-Net in Base 4 — Upper bound on s
There is no (27, 36, 136757)-net in base 4, because
- 1 times m-reduction [i] would yield (27, 35, 136757)-net in base 4, but
- the generalized Rao bound for nets shows that 4m ≥ 1180 614270 248005 249429 > 435 [i]