Best Known (20, 20+29, s)-Nets in Base 128
(20, 20+29, 384)-Net over F128 — Constructive and digital
Digital (20, 49, 384)-net over F128, using
- (u, u+v)-construction [i] based on
- digital (3, 17, 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, 32, 192)-net over F128, using
- net from sequence [i] based on digital (3, 191)-sequence over F128 (see above)
- digital (3, 17, 192)-net over F128, using
(20, 20+29, 407)-Net in Base 128 — Constructive
(20, 49, 407)-net in base 128, using
- (u, u+v)-construction [i] based on
- digital (1, 15, 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, 34, 257)-net in base 128, using
- 6 times m-reduction [i] based on (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
- 6 times m-reduction [i] based on (5, 40, 257)-net in base 128, using
- digital (1, 15, 150)-net over F128, using
(20, 20+29, 451)-Net over F128 — Digital
Digital (20, 49, 451)-net over F128, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(12849, 451, F128, 29) (dual of [451, 402, 30]-code), using
- 60 step Varšamov–Edel lengthening with (ri) = (3, 0, 1, 12 times 0, 1, 44 times 0) [i] based on linear OA(12844, 386, F128, 29) (dual of [386, 342, 30]-code), using
- extended algebraic-geometric code AGe(F,356P) [i] based on function field F/F128 with g(F) = 15 and N(F) ≥ 386, using
- 60 step Varšamov–Edel lengthening with (ri) = (3, 0, 1, 12 times 0, 1, 44 times 0) [i] based on linear OA(12844, 386, F128, 29) (dual of [386, 342, 30]-code), using
(20, 20+29, 513)-Net in Base 128
(20, 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
(20, 20+29, 798675)-Net in Base 128 — Upper bound on s
There is no (20, 49, 798676)-net in base 128, because
- 1 times m-reduction [i] would yield (20, 48, 798676)-net in base 128, but
- the generalized Rao bound for nets shows that 128m ≥ 139985 464223 345368 217791 555835 186749 173412 360277 943927 117873 836152 125998 790470 546621 314895 615956 217244 > 12848 [i]