Best Known (115, 115+31, s)-Nets in Base 4
(115, 115+31, 1049)-Net over F4 — Constructive and digital
Digital (115, 146, 1049)-net over F4, using
- (u, u+v)-construction [i] based on
- digital (7, 22, 21)-net over F4, using
- net from sequence [i] based on digital (7, 20)-sequence over F4, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F4 with g(F) = 7 and N(F) ≥ 21, using
- net from sequence [i] based on digital (7, 20)-sequence over F4, using
- digital (93, 124, 1028)-net over F4, using
- trace code for nets [i] based on digital (0, 31, 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, 31, 257)-net over F256, using
- digital (7, 22, 21)-net over F4, using
(115, 115+31, 3961)-Net over F4 — Digital
Digital (115, 146, 3961)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4146, 3961, F4, 31) (dual of [3961, 3815, 32]-code), using
- discarding factors / shortening the dual code based on linear OA(4146, 4110, F4, 31) (dual of [4110, 3964, 32]-code), using
- construction X applied to C([0,16]) ⊂ C([0,14]) [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]
- linear OA(4133, 4097, F4, 29) (dual of [4097, 3964, 30]-code), using the expurgated narrow-sense BCH-code C(I) with length 4097 | 412−1, defining interval I = [0,14], and minimum distance d ≥ |{−14,−13,…,14}|+1 = 30 (BCH-bound) [i]
- linear OA(41, 13, F4, 1) (dual of [13, 12, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(41, s, F4, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to C([0,16]) ⊂ C([0,14]) [i] based on
- discarding factors / shortening the dual code based on linear OA(4146, 4110, F4, 31) (dual of [4110, 3964, 32]-code), using
(115, 115+31, 1414343)-Net in Base 4 — Upper bound on s
There is no (115, 146, 1414344)-net in base 4, because
- 1 times m-reduction [i] would yield (115, 145, 1414344)-net in base 4, but
- the generalized Rao bound for nets shows that 4m ≥ 1989 299928 351977 934673 853313 628322 419061 334151 532276 102970 066397 557962 195736 738987 092872 > 4145 [i]