MinT - Best Known (12, 12+72, s)-Nets in Base 64

Best Known (12, 12+72, s)-Nets in Base 64

(12, 12+72, 177)-Net over F₆₄ — Constructive and digital

Digital (12, 84, 177)-net over F₆₄, using

t-expansion [i] based on digital (7, 84, 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]

(12, 12+72, 257)-Net over F₆₄ — Digital

Digital (12, 84, 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]

(12, 12+72, 3695)-Net in Base 64 — Upper bound on s

There is no (12, 84, 3696)-net in base 64, because

the generalized Rao bound for nets shows that 64^mÂ â‰¥ 52 512302 376760 546988 554364 627081 501354 412025 413937 886790 134209 193031 103244 773398 646065 772304 814239 672814 529782 557578 486133 081793 460968 707269 962660 900895 > 64⁸⁴ [i]