Best Known (31, 31+9, s)-Nets in Base 4
(31, 31+9, 1033)-Net over F4 — Constructive and digital
Digital (31, 40, 1033)-net over F4, using
- (u, u+v)-construction [i] based on
- digital (0, 4, 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 (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
- trace code for nets [i] based on digital (0, 9, 257)-net over F256, using
- digital (0, 4, 5)-net over F4, using
(31, 31+9, 2543)-Net over F4 — Digital
Digital (31, 40, 2543)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(440, 2543, F4, 9) (dual of [2543, 2503, 10]-code), using
- discarding factors / shortening the dual code based on linear OA(440, 4103, F4, 9) (dual of [4103, 4063, 10]-code), using
- construction X applied to C([0,4]) ⊂ C([0,2]) [i] based on
- linear OA(437, 4097, F4, 9) (dual of [4097, 4060, 10]-code), using the expurgated narrow-sense BCH-code C(I) with length 4097 | 412−1, defining interval I = [0,4], and minimum distance d ≥ |{−4,−3,…,4}|+1 = 10 (BCH-bound) [i]
- linear OA(425, 4097, F4, 5) (dual of [4097, 4072, 6]-code), using the expurgated narrow-sense BCH-code C(I) with length 4097 | 412−1, defining interval I = [0,2], and minimum distance d ≥ |{−4,−2,0,2,4}|+1 = 6 (BCH-bound) [i]
- linear OA(43, 6, F4, 3) (dual of [6, 3, 4]-code or 6-arc in PG(2,4) or 6-cap in PG(2,4)), using
- construction X applied to C([0,4]) ⊂ C([0,2]) [i] based on
- discarding factors / shortening the dual code based on linear OA(440, 4103, F4, 9) (dual of [4103, 4063, 10]-code), using
(31, 31+9, 547033)-Net in Base 4 — Upper bound on s
There is no (31, 40, 547034)-net in base 4, because
- 1 times m-reduction [i] would yield (31, 39, 547034)-net in base 4, but
- the generalized Rao bound for nets shows that 4m ≥ 302231 728456 995460 142554 > 439 [i]