MinT - Best Known (66−41, 66, s)-Nets in Base 256

Best Known (66−41, 66, s)-Nets in Base 256

(66−41, 66, 519)-Net over F₂₅₆ — Constructive and digital

Digital (25, 66, 519)-net over F₂₅₆, using

(u,Â u+v)-construction [i] based on
1. digital (2, 22, 259)-net over F₂₅₆, using
  - net from sequence [i] based on digital (2, 258)-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, using
    - Niederreiter sequence [i]

(66−41, 66, 1025)-Net over F₂₅₆ — Digital

Digital (25, 66, 1025)-net over F₂₅₆, using

t-expansion [i] based on digital (24, 66, 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]

(66−41, 66, 2185465)-Net in Base 256 — Upper bound on s

There is no (25, 66, 2185466)-net in base 256, because

1 times m-reduction [i] would yield (25, 65, 2185466)-net in base 256, but
- the generalized Rao bound for nets shows that 256^mÂ â‰¥ 3 432402 016664 059866 491954 191195 141322 486062 507414 057266 918995 901003 906271 846653 512727 518176 449856 050607 847729 491135 784244 773924 770716 645359 054162 260213 780726 > 256⁶⁵ [i]