Best Known (125, 125+33, s)-Nets in Base 4
(125, 125+33, 1055)-Net over F4 — Constructive and digital
Digital (125, 158, 1055)-net over F4, using
- (u, u+v)-construction [i] based on
- digital (10, 26, 27)-net over F4, using
- net from sequence [i] based on digital (10, 26)-sequence over F4, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F4 with g(F) = 10 and N(F) ≥ 27, using
- net from sequence [i] based on digital (10, 26)-sequence over F4, 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 (10, 26, 27)-net over F4, using
(125, 125+33, 4296)-Net over F4 — Digital
Digital (125, 158, 4296)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4158, 4296, F4, 33) (dual of [4296, 4138, 34]-code), using
- 186 step Varšamov–Edel lengthening with (ri) = (4, 1, 1, 0, 0, 1, 4 times 0, 1, 6 times 0, 1, 12 times 0, 1, 18 times 0, 1, 29 times 0, 1, 43 times 0, 1, 62 times 0) [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]
- 186 step Varšamov–Edel lengthening with (ri) = (4, 1, 1, 0, 0, 1, 4 times 0, 1, 6 times 0, 1, 12 times 0, 1, 18 times 0, 1, 29 times 0, 1, 43 times 0, 1, 62 times 0) [i] based on linear OA(4145, 4097, F4, 33) (dual of [4097, 3952, 34]-code), using
(125, 125+33, 1832854)-Net in Base 4 — Upper bound on s
There is no (125, 158, 1832855)-net in base 4, because
- 1 times m-reduction [i] would yield (125, 157, 1832855)-net in base 4, but
- the generalized Rao bound for nets shows that 4m ≥ 33374 815449 801876 666043 940618 456277 949598 166469 215195 864608 195680 827216 164664 351633 940405 239229 > 4157 [i]