Best Known (58−35, 58, s)-Nets in Base 128
(58−35, 58, 384)-Net over F128 — Constructive and digital
Digital (23, 58, 384)-net over F128, using
- (u, u+v)-construction [i] based on
- digital (3, 20, 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, 38, 192)-net over F128, using
- net from sequence [i] based on digital (3, 191)-sequence over F128 (see above)
- digital (3, 20, 192)-net over F128, using
(58−35, 58, 407)-Net in Base 128 — Constructive
(23, 58, 407)-net in base 128, 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
- (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
- digital (1, 18, 150)-net over F128, using
(58−35, 58, 438)-Net over F128 — Digital
Digital (23, 58, 438)-net over F128, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(12858, 438, F128, 35) (dual of [438, 380, 36]-code), using
- discarding factors / shortening the dual code based on linear OA(12858, 439, F128, 35) (dual of [439, 381, 36]-code), using
- 77 step Varšamov–Edel lengthening with (ri) = (6, 0, 0, 1, 9 times 0, 1, 21 times 0, 1, 41 times 0) [i] based on linear OA(12849, 353, F128, 35) (dual of [353, 304, 36]-code), using
- extended algebraic-geometric code AGe(F,317P) [i] based on function field F/F128 with g(F) = 14 and N(F) ≥ 353, using
- 77 step Varšamov–Edel lengthening with (ri) = (6, 0, 0, 1, 9 times 0, 1, 21 times 0, 1, 41 times 0) [i] based on linear OA(12849, 353, F128, 35) (dual of [353, 304, 36]-code), using
- discarding factors / shortening the dual code based on linear OA(12858, 439, F128, 35) (dual of [439, 381, 36]-code), using
(58−35, 58, 513)-Net in Base 128
(23, 58, 513)-net in base 128, using
- t-expansion [i] based on (17, 58, 513)-net in base 128, using
- 14 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
- 14 times m-reduction [i] based on (17, 72, 513)-net in base 128, using
(58−35, 58, 656881)-Net in Base 128 — Upper bound on s
There is no (23, 58, 656882)-net in base 128, because
- 1 times m-reduction [i] would yield (23, 57, 656882)-net in base 128, but
- the generalized Rao bound for nets shows that 128m ≥ 1 291133 960966 505151 279380 609767 958516 894640 973118 437567 052884 233135 213507 210215 217829 558185 490036 199696 154931 147374 213661 > 12857 [i]