Best Known (127, 127+32, s)-Nets in Base 4
(127, 127+32, 1061)-Net over F4 — Constructive and digital
Digital (127, 159, 1061)-net over F4, using
- (u, u+v)-construction [i] based on
- digital (15, 31, 33)-net over F4, using
- net from sequence [i] based on digital (15, 32)-sequence over F4, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F4 with g(F) = 15 and N(F) ≥ 33, using
- net from sequence [i] based on digital (15, 32)-sequence over F4, using
- digital (96, 128, 1028)-net over F4, using
- trace code for nets [i] based on digital (0, 32, 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, 32, 257)-net over F256, using
- digital (15, 31, 33)-net over F4, using
(127, 127+32, 5090)-Net over F4 — Digital
Digital (127, 159, 5090)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4159, 5090, F4, 32) (dual of [5090, 4931, 33]-code), using
- 979 step Varšamov–Edel lengthening with (ri) = (3, 1, 0, 0, 1, 5 times 0, 1, 9 times 0, 1, 17 times 0, 1, 29 times 0, 1, 46 times 0, 1, 68 times 0, 1, 98 times 0, 1, 130 times 0, 1, 163 times 0, 1, 189 times 0, 1, 210 times 0) [i] based on linear OA(4144, 4096, F4, 32) (dual of [4096, 3952, 33]-code), using
- 1 times truncation [i] based on linear OA(4145, 4097, F4, 33) (dual of [4097, 3952, 34]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 4097 | 412−1, defining interval I = [0,16], and minimum distance d ≥ |{−16,−15,…,16}|+1 = 34 (BCH-bound) [i]
- 1 times truncation [i] based on linear OA(4145, 4097, F4, 33) (dual of [4097, 3952, 34]-code), using
- 979 step Varšamov–Edel lengthening with (ri) = (3, 1, 0, 0, 1, 5 times 0, 1, 9 times 0, 1, 17 times 0, 1, 29 times 0, 1, 46 times 0, 1, 68 times 0, 1, 98 times 0, 1, 130 times 0, 1, 163 times 0, 1, 189 times 0, 1, 210 times 0) [i] based on linear OA(4144, 4096, F4, 32) (dual of [4096, 3952, 33]-code), using
(127, 127+32, 2179646)-Net in Base 4 — Upper bound on s
There is no (127, 159, 2179647)-net in base 4, because
- the generalized Rao bound for nets shows that 4m ≥ 533998 478282 903718 992097 639569 453155 109075 496349 869707 776474 694813 364670 263232 304979 940147 027717 > 4159 [i]