MinT - Best Known (67−41, 67, s)-Nets in Base 256

Best Known (67−41, 67, s)-Nets in Base 256

(67−41, 67, 520)-Net over F₂₅₆ — Constructive and digital

Digital (26, 67, 520)-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 (3, 44, 260)-net over F₂₅₆, using
  - net from sequence [i] based on digital (3, 259)-sequence over F₂₅₆ (see above)

(67−41, 67, 1025)-Net over F₂₅₆ — Digital

Digital (26, 67, 1025)-net over F₂₅₆, using

t-expansion [i] based on digital (24, 67, 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]

(67−41, 67, 2883742)-Net in Base 256 — Upper bound on s

There is no (26, 67, 2883743)-net in base 256, because

1 times m-reduction [i] would yield (26, 66, 2883743)-net in base 256, but
- the generalized Rao bound for nets shows that 256^mÂ â‰¥ 878 696524 470068 280125 247686 624925 873601 371899 338369 416156 487305 785092 497059 023687 907122 010993 078512 138382 261760 068871 086102 582098 270789 548788 051905 962422 351551 > 256⁶⁶ [i]