Best Known (59−37, 59, s)-Nets in Base 128
(59−37, 59, 342)-Net over F128 — Constructive and digital
Digital (22, 59, 342)-net over F128, using
- (u, u+v)-construction [i] based on
- digital (1, 19, 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, 40, 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, 19, 150)-net over F128, using
(59−37, 59, 388)-Net over F128 — Digital
Digital (22, 59, 388)-net over F128, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(12859, 388, F128, 4, 37) (dual of [(388, 4), 1493, 38]-NRT-code), using
- strength reduction [i] based on linear OOA(12859, 388, F128, 4, 38) (dual of [(388, 4), 1493, 39]-NRT-code), using
- construction X applied to AG(4;F,1501P) ⊂ AG(4;F,1508P) [i] based on
- linear OOA(12853, 385, F128, 4, 38) (dual of [(385, 4), 1487, 39]-NRT-code), using algebraic-geometric NRT-code AG(4;F,1501P) [i] based on function field F/F128 with g(F) = 15 and N(F) ≥ 386, using
- linear OOA(12846, 385, F128, 4, 31) (dual of [(385, 4), 1494, 32]-NRT-code), using algebraic-geometric NRT-code AG(4;F,1508P) [i] based on function field F/F128 with g(F) = 15 and N(F) ≥ 386 (see above)
- linear OOA(1286, 3, F128, 4, 6) (dual of [(3, 4), 6, 7]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(1286, 128, F128, 4, 6) (dual of [(128, 4), 506, 7]-NRT-code), using
- Reed–Solomon NRT-code RS(4;506,128) [i]
- discarding factors / shortening the dual code based on linear OOA(1286, 128, F128, 4, 6) (dual of [(128, 4), 506, 7]-NRT-code), using
- construction X applied to AG(4;F,1501P) ⊂ AG(4;F,1508P) [i] based on
- strength reduction [i] based on linear OOA(12859, 388, F128, 4, 38) (dual of [(388, 4), 1493, 39]-NRT-code), using
(59−37, 59, 513)-Net in Base 128
(22, 59, 513)-net in base 128, using
- t-expansion [i] based on (17, 59, 513)-net in base 128, using
- 13 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
- 13 times m-reduction [i] based on (17, 72, 513)-net in base 128, using
(59−37, 59, 366618)-Net in Base 128 — Upper bound on s
There is no (22, 59, 366619)-net in base 128, because
- 1 times m-reduction [i] would yield (22, 58, 366619)-net in base 128, but
- the generalized Rao bound for nets shows that 128m ≥ 165 264529 515502 001692 465109 789365 906468 970670 337709 459980 082141 603181 155483 813081 422251 615797 312807 609029 820793 119095 906595 > 12858 [i]