MinT - Best Known (86−78, 86, s)-Nets in Base 64

Best Known (86−78, 86, s)-Nets in Base 64

(86−78, 86, 177)-Net over F₆₄ — Constructive and digital

Digital (8, 86, 177)-net over F₆₄, using

t-expansion [i] based on digital (7, 86, 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]

(86−78, 86, 2321)-Net in Base 64 — Upper bound on s

There is no (8, 86, 2322)-net in base 64, because

6 times m-reduction [i] would yield (8, 80, 2322)-net in base 64, but
- the generalized Rao bound for nets shows that 64^mÂ â‰¥ 3 146225 264235 909920 729481 449982 968435 369868 272243 696932 507446 836547 548541 584737 869342 232201 208624 793712 046242 248383 564625 633447 509112 999547 893280 > 64⁸⁰ [i]