Best Known (55−34, 55, s)-Nets in Base 128
(55−34, 55, 342)-Net over F128 — Constructive and digital
Digital (21, 55, 342)-net over F128, using
- (u, u+v)-construction [i] based on
- digital (1, 18, 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, 37, 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, 18, 150)-net over F128, using
(55−34, 55, 389)-Net over F128 — Digital
Digital (21, 55, 389)-net over F128, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(12855, 389, F128, 3, 34) (dual of [(389, 3), 1112, 35]-NRT-code), using
- construction X applied to AG(3;F,1120P) ⊂ AG(3;F,1127P) [i] based on
- linear OOA(12849, 385, F128, 3, 34) (dual of [(385, 3), 1106, 35]-NRT-code), using algebraic-geometric NRT-code AG(3;F,1120P) [i] based on function field F/F128 with g(F) = 15 and N(F) ≥ 386, using
- linear OOA(12842, 385, F128, 3, 27) (dual of [(385, 3), 1113, 28]-NRT-code), using algebraic-geometric NRT-code AG(3;F,1127P) [i] based on function field F/F128 with g(F) = 15 and N(F) ≥ 386 (see above)
- linear OOA(1286, 4, F128, 3, 6) (dual of [(4, 3), 6, 7]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(1286, 128, F128, 3, 6) (dual of [(128, 3), 378, 7]-NRT-code), using
- Reed–Solomon NRT-code RS(3;378,128) [i]
- discarding factors / shortening the dual code based on linear OOA(1286, 128, F128, 3, 6) (dual of [(128, 3), 378, 7]-NRT-code), using
- construction X applied to AG(3;F,1120P) ⊂ AG(3;F,1127P) [i] based on
(55−34, 55, 513)-Net in Base 128
(21, 55, 513)-net in base 128, using
- t-expansion [i] based on (17, 55, 513)-net in base 128, using
- 17 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
- 17 times m-reduction [i] based on (17, 72, 513)-net in base 128, using
(55−34, 55, 371172)-Net in Base 128 — Upper bound on s
There is no (21, 55, 371173)-net in base 128, because
- the generalized Rao bound for nets shows that 128m ≥ 78 805211 657950 232793 059144 753650 370264 340910 933676 395077 354669 104323 259519 276153 866056 534193 000985 803976 452326 877912 > 12855 [i]