MinT - Best Known (48−36, 48, s)-Nets in Base 64

Best Known (48−36, 48, s)-Nets in Base 64

(48−36, 48, 177)-Net over F₆₄ — Constructive and digital

Digital (12, 48, 177)-net over F₆₄, using

t-expansion [i] based on digital (7, 48, 177)-net over F₆₄, using
- net from sequence [i] based on digital (7, 176)-sequence over F₆₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 7 and N(F) ≥ 177, using
    - a function field by GarcÃa and Quoos [i]

(48−36, 48, 257)-Net in Base 64 — Constructive

(12, 48, 257)-net in base 64, using

base change [i] based on digital (0, 36, 257)-net over F₂₅₆, using
- net from sequence [i] based on digital (0, 256)-sequence over F₂₅₆, using
  - generalized Faure sequence [i]
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₅₆ with g(F) = 0 and N(F) ≥ 257, using
    - the rational function field F₂₅₆(x) [i]
  - Niederreiter sequence [i]

(48−36, 48, 257)-Net over F₆₄ — Digital

Digital (12, 48, 257)-net over F₆₄, using

net from sequence [i] based on digital (12, 256)-sequence over F₆₄, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 12 and N(F) ≥ 257, using
  - K_1,2 from the tower of function fields by Niederreiter and Xing based on the tower by GarcÃa and Stichtenoth over F₆₄ [i]

(48−36, 48, 7848)-Net in Base 64 — Upper bound on s

There is no (12, 48, 7849)-net in base 64, because

the generalized Rao bound for nets shows that 64^mÂ â‰¥ 497 955020 807827 067272 782260 513643 463331 019955 152530 447917 173960 713198 012473 534853 418952 > 64⁴⁸ [i]