MinT - Best Known (72−59, 72, s)-Nets in Base 64

Best Known (72−59, 72, s)-Nets in Base 64

(72−59, 72, 177)-Net over F₆₄ — Constructive and digital

Digital (13, 72, 177)-net over F₆₄, using

t-expansion [i] based on digital (7, 72, 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]

(72−59, 72, 257)-Net over F₆₄ — Digital

Digital (13, 72, 257)-net over F₆₄, using

t-expansion [i] based on digital (12, 72, 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]

(72−59, 72, 4881)-Net in Base 64 — Upper bound on s

There is no (13, 72, 4882)-net in base 64, because

1 times m-reduction [i] would yield (13, 71, 4882)-net in base 64, but
- the generalized Rao bound for nets shows that 64^mÂ â‰¥ 174 140678 343692 273369 950661 383656 305756 129297 026354 758355 299843 856015 107253 772155 327465 825671 127211 698158 475102 455517 243817 646016 > 64⁷¹ [i]