Best Known (19, 19+30, s)-Nets in Base 128
(19, 19+30, 342)-Net over F128 — Constructive and digital
Digital (19, 49, 342)-net over F128, 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
- digital (3, 33, 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 (1, 16, 150)-net over F128, using
(19, 19+30, 388)-Net over F128 — Digital
Digital (19, 49, 388)-net over F128, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(12849, 388, F128, 2, 30) (dual of [(388, 2), 727, 31]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(12849, 389, F128, 2, 30) (dual of [(389, 2), 729, 31]-NRT-code), using
- construction X applied to AG(2;F,739P) ⊂ AG(2;F,744P) [i] based on
- linear OOA(12845, 385, F128, 2, 30) (dual of [(385, 2), 725, 31]-NRT-code), using algebraic-geometric NRT-code AG(2;F,739P) [i] based on function field F/F128 with g(F) = 15 and N(F) ≥ 386, using
- linear OOA(12840, 385, F128, 2, 25) (dual of [(385, 2), 730, 26]-NRT-code), using algebraic-geometric NRT-code AG(2;F,744P) [i] based on function field F/F128 with g(F) = 15 and N(F) ≥ 386 (see above)
- linear OOA(1284, 4, F128, 2, 4) (dual of [(4, 2), 4, 5]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(1284, 128, F128, 2, 4) (dual of [(128, 2), 252, 5]-NRT-code), using
- Reed–Solomon NRT-code RS(2;252,128) [i]
- discarding factors / shortening the dual code based on linear OOA(1284, 128, F128, 2, 4) (dual of [(128, 2), 252, 5]-NRT-code), using
- construction X applied to AG(2;F,739P) ⊂ AG(2;F,744P) [i] based on
- discarding factors / shortening the dual code based on linear OOA(12849, 389, F128, 2, 30) (dual of [(389, 2), 729, 31]-NRT-code), using
(19, 19+30, 513)-Net in Base 128
(19, 49, 513)-net in base 128, using
- t-expansion [i] based on (17, 49, 513)-net in base 128, using
- 23 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
- 23 times m-reduction [i] based on (17, 72, 513)-net in base 128, using
(19, 19+30, 386818)-Net in Base 128 — Upper bound on s
There is no (19, 49, 386819)-net in base 128, because
- the generalized Rao bound for nets shows that 128m ≥ 17 918144 956637 549383 738977 188890 970973 525202 696469 784630 831335 357569 446483 221520 537248 654613 778369 393920 > 12849 [i]