MinT - Best Known (12, 12+79, s)-Nets in Base 64

Best Known (12, 12+79, s)-Nets in Base 64

(12, 12+79, 177)-Net over F₆₄ — Constructive and digital

Digital (12, 91, 177)-net over F₆₄, using

t-expansion [i] based on digital (7, 91, 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+79, 257)-Net over F₆₄ — Digital

Digital (12, 91, 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+79, 3579)-Net in Base 64 — Upper bound on s

There is no (12, 91, 3580)-net in base 64, because

1 times m-reduction [i] would yield (12, 90, 3580)-net in base 64, but
- the generalized Rao bound for nets shows that 64^mÂ â‰¥ 3 619121 033637 094786 326891 337075 356998 238691 059224 891667 718863 596977 133980 229845 186198 815004 349494 962918 202317 985458 620479 107469 309966 119493 925230 792495 785620 259058 > 64⁹⁰ [i]