MinT - Best Known (57−33, 57, s)-Nets in Base 256

Best Known (57−33, 57, s)-Nets in Base 256

(57−33, 57, 522)-Net over F₂₅₆ — Constructive and digital

Digital (24, 57, 522)-net over F₂₅₆, using

(u,Â u+v)-construction [i] based on
1. digital (4, 20, 261)-net over F₂₅₆, using
  - net from sequence [i] based on digital (4, 260)-sequence over F₂₅₆, using
    - Niederreiter sequence [i]
2. digital (4, 37, 261)-net over F₂₅₆, using
  - net from sequence [i] based on digital (4, 260)-sequence over F₂₅₆ (see above)

(57−33, 57, 1025)-Net over F₂₅₆ — Digital

Digital (24, 57, 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]

(57−33, 57, 7158762)-Net in Base 256 — Upper bound on s

There is no (24, 57, 7158763)-net in base 256, because

1 times m-reduction [i] would yield (24, 56, 7158763)-net in base 256, but
- the generalized Rao bound for nets shows that 256^mÂ â‰¥ 726 839281 682463 472770 681731 308566 785797 998532 086875 688997 467107 821997 756801 409646 926031 756947 348758 377694 639434 075957 859460 242887 329166 > 256⁵⁶ [i]