Best Known (51−30, 51, s)-Nets in Base 128
(51−30, 51, 384)-Net over F128 — Constructive and digital
Digital (21, 51, 384)-net over F128, using
- (u, u+v)-construction [i] based on
- digital (3, 18, 192)-net over F128, using
- net from sequence [i] based on digital (3, 191)-sequence over F128, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F128 with g(F) = 3 and N(F) ≥ 192, using
- net from sequence [i] based on digital (3, 191)-sequence over F128, using
- digital (3, 33, 192)-net over F128, using
- net from sequence [i] based on digital (3, 191)-sequence over F128 (see above)
- digital (3, 18, 192)-net over F128, using
(51−30, 51, 407)-Net in Base 128 — Constructive
(21, 51, 407)-net in base 128, using
- (u, u+v)-construction [i] based on
- digital (1, 16, 150)-net over F128, using
- net from sequence [i] based on digital (1, 149)-sequence over F128, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F128 with g(F) = 1 and N(F) ≥ 150, using
- net from sequence [i] based on digital (1, 149)-sequence over F128, using
- (5, 35, 257)-net in base 128, using
- 5 times m-reduction [i] based on (5, 40, 257)-net in base 128, using
- base change [i] based on digital (0, 35, 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
- base change [i] based on digital (0, 35, 257)-net over F256, using
- 5 times m-reduction [i] based on (5, 40, 257)-net in base 128, using
- digital (1, 16, 150)-net over F128, using
(51−30, 51, 483)-Net over F128 — Digital
Digital (21, 51, 483)-net over F128, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(12851, 483, F128, 30) (dual of [483, 432, 31]-code), using
- 91 step Varšamov–Edel lengthening with (ri) = (4, 5 times 0, 1, 24 times 0, 1, 59 times 0) [i] based on linear OA(12845, 386, F128, 30) (dual of [386, 341, 31]-code), using
- extended algebraic-geometric code AGe(F,355P) [i] based on function field F/F128 with g(F) = 15 and N(F) ≥ 386, using
- 91 step Varšamov–Edel lengthening with (ri) = (4, 5 times 0, 1, 24 times 0, 1, 59 times 0) [i] based on linear OA(12845, 386, F128, 30) (dual of [386, 341, 31]-code), using
(51−30, 51, 513)-Net in Base 128
(21, 51, 513)-net in base 128, using
- t-expansion [i] based on (17, 51, 513)-net in base 128, using
- 21 times m-reduction [i] based on (17, 72, 513)-net in base 128, using
- base change [i] based on digital (8, 63, 513)-net over F256, using
- net from sequence [i] based on digital (8, 512)-sequence over F256, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F256 with g(F) = 8 and N(F) ≥ 513, using
- K1,1 from the tower of function fields by Niederreiter and Xing based on the tower by GarcÃa and Stichtenoth over F256 [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F256 with g(F) = 8 and N(F) ≥ 513, using
- net from sequence [i] based on digital (8, 512)-sequence over F256, using
- base change [i] based on digital (8, 63, 513)-net over F256, using
- 21 times m-reduction [i] based on (17, 72, 513)-net in base 128, using
(51−30, 51, 738707)-Net in Base 128 — Upper bound on s
There is no (21, 51, 738708)-net in base 128, because
- the generalized Rao bound for nets shows that 128m ≥ 293569 551952 191159 995188 024104 547442 881401 534441 366636 767364 830459 698080 823500 601544 444538 990849 717119 298876 > 12851 [i]