Best Known (126, 126+33, s)-Nets in Base 4
(126, 126+33, 1055)-Net over F4 — Constructive and digital
Digital (126, 159, 1055)-net over F4, using
- 41 times duplication [i] based on 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
- (u, u+v)-construction [i] based on
(126, 126+33, 4383)-Net over F4 — Digital
Digital (126, 159, 4383)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4159, 4383, F4, 33) (dual of [4383, 4224, 34]-code), using
- 272 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, 1, 85 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]
- 272 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, 1, 85 times 0) [i] based on linear OA(4145, 4097, F4, 33) (dual of [4097, 3952, 34]-code), using
(126, 126+33, 1998743)-Net in Base 4 — Upper bound on s
There is no (126, 159, 1998744)-net in base 4, because
- 1 times m-reduction [i] would yield (126, 158, 1998744)-net in base 4, but
- the generalized Rao bound for nets shows that 4m ≥ 133499 569718 497555 170533 670426 524085 922815 255704 886889 814370 753950 080647 715590 773188 136833 877101 > 4158 [i]