MinT - Best Known (27, 27+41, s)-Nets in Base 256

Best Known (27, 27+41, s)-Nets in Base 256

(27, 27+41, 521)-Net over F₂₅₆ — Constructive and digital

Digital (27, 68, 521)-net over F₂₅₆, using

(u,Â u+v)-construction [i] based on
1. digital (3, 23, 260)-net over F₂₅₆, using
  - net from sequence [i] based on digital (3, 259)-sequence over F₂₅₆, using
    - Niederreiter sequence [i]
2. digital (4, 45, 261)-net over F₂₅₆, using
  - net from sequence [i] based on digital (4, 260)-sequence over F₂₅₆, using
    - Niederreiter sequence [i]

(27, 27+41, 1025)-Net over F₂₅₆ — Digital

Digital (27, 68, 1025)-net over F₂₅₆, using

t-expansion [i] based on digital (24, 68, 1025)-net over F₂₅₆, using
- net from sequence [i] based on digital (24, 1024)-sequence over F₂₅₆, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₅₆ with g(F) = 24 and N(F) ≥ 1025, using
    - K_1,2 from the tower of function fields by Niederreiter and Xing based on the tower by GarcÃa and Stichtenoth over F₂₅₆ [i]

(27, 27+41, 3805124)-Net in Base 256 — Upper bound on s

There is no (27, 68, 3805125)-net in base 256, because

1 times m-reduction [i] would yield (27, 67, 3805125)-net in base 256, but
- the generalized Rao bound for nets shows that 256^mÂ â‰¥ 224946 592230 016320 261772 424132 252418 420897 397703 736695 870599 926026 808512 853790 805717 207315 887547 000649 874398 857686 514513 515241 733842 746718 443629 574902 943813 621876 > 256⁶⁷ [i]