Best Known (60−37, 60, s)-Nets in Base 128
(60−37, 60, 345)-Net over F128 — Constructive and digital
Digital (23, 60, 345)-net over F128, using
- (u, u+v)-construction [i] based on
- digital (0, 18, 129)-net over F128, using
- net from sequence [i] based on digital (0, 128)-sequence over F128, using
- generalized Faure sequence [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F128 with g(F) = 0 and N(F) ≥ 129, using
- the rational function field F128(x) [i]
- Niederreiter sequence [i]
- net from sequence [i] based on digital (0, 128)-sequence over F128, using
- digital (5, 42, 216)-net over F128, using
- net from sequence [i] based on digital (5, 215)-sequence over F128, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F128 with g(F) = 5 and N(F) ≥ 216, using
- net from sequence [i] based on digital (5, 215)-sequence over F128, using
- digital (0, 18, 129)-net over F128, using
(60−37, 60, 393)-Net over F128 — Digital
Digital (23, 60, 393)-net over F128, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(12860, 393, F128, 2, 37) (dual of [(393, 2), 726, 38]-NRT-code), using
- construction X applied to AG(2;F,732P) ⊂ AG(2;F,741P) [i] based on
- linear OOA(12852, 385, F128, 2, 37) (dual of [(385, 2), 718, 38]-NRT-code), using algebraic-geometric NRT-code AG(2;F,732P) [i] based on function field F/F128 with g(F) = 15 and N(F) ≥ 386, using
- linear OOA(12843, 385, F128, 2, 28) (dual of [(385, 2), 727, 29]-NRT-code), using algebraic-geometric NRT-code AG(2;F,741P) [i] based on function field F/F128 with g(F) = 15 and N(F) ≥ 386 (see above)
- linear OOA(1288, 8, F128, 2, 8) (dual of [(8, 2), 8, 9]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(1288, 128, F128, 2, 8) (dual of [(128, 2), 248, 9]-NRT-code), using
- Reed–Solomon NRT-code RS(2;248,128) [i]
- discarding factors / shortening the dual code based on linear OOA(1288, 128, F128, 2, 8) (dual of [(128, 2), 248, 9]-NRT-code), using
- construction X applied to AG(2;F,732P) ⊂ AG(2;F,741P) [i] based on
(60−37, 60, 513)-Net in Base 128
(23, 60, 513)-net in base 128, using
- t-expansion [i] based on (17, 60, 513)-net in base 128, using
- 12 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
- 12 times m-reduction [i] based on (17, 72, 513)-net in base 128, using
(60−37, 60, 480047)-Net in Base 128 — Upper bound on s
There is no (23, 60, 480048)-net in base 128, because
- 1 times m-reduction [i] would yield (23, 59, 480048)-net in base 128, but
- the generalized Rao bound for nets shows that 128m ≥ 21153 909618 929235 526649 753709 694473 324830 098334 894695 403727 335531 824417 399690 765791 624807 467343 774768 962548 096616 575566 504746 > 12859 [i]