Best Known (72−51, 72, s)-Nets in Base 128
(72−51, 72, 288)-Net over F128 — Constructive and digital
Digital (21, 72, 288)-net over F128, using
- t-expansion [i] based on digital (9, 72, 288)-net over F128, using
- net from sequence [i] based on digital (9, 287)-sequence over F128, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F128 with g(F) = 9 and N(F) ≥ 288, using
- net from sequence [i] based on digital (9, 287)-sequence over F128, using
(72−51, 72, 386)-Net over F128 — Digital
Digital (21, 72, 386)-net over F128, using
- t-expansion [i] based on digital (15, 72, 386)-net over F128, using
- net from sequence [i] based on digital (15, 385)-sequence over F128, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F128 with g(F) = 15 and N(F) ≥ 386, using
- net from sequence [i] based on digital (15, 385)-sequence over F128, using
(72−51, 72, 513)-Net in Base 128
(21, 72, 513)-net in base 128, using
- t-expansion [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
(72−51, 72, 77308)-Net in Base 128 — Upper bound on s
There is no (21, 72, 77309)-net in base 128, because
- 1 times m-reduction [i] would yield (21, 71, 77309)-net in base 128, but
- the generalized Rao bound for nets shows that 128m ≥ 409177 259436 370742 916995 362892 103058 197523 470790 805854 522228 667427 915912 556996 025576 530226 657093 313143 847003 575261 818450 191517 270216 731492 484852 658892 > 12871 [i]