MinT - Best Known (90−51, 90, s)-Nets in Base 64

Best Known (90−51, 90, s)-Nets in Base 64

(90−51, 90, 513)-Net over F₆₄ — Constructive and digital

Digital (39, 90, 513)-net over F₆₄, using

t-expansion [i] based on digital (28, 90, 513)-net over F₆₄, using
- net from sequence [i] based on digital (28, 512)-sequence over F₆₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 28 and N(F) ≥ 513, using
    - the Hermitian function field over F₆₄ [i]

(90−51, 90, 558)-Net over F₆₄ — Digital

Digital (39, 90, 558)-net over F₆₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(90−51, 90, 434784)-Net in Base 64 — Upper bound on s

There is no (39, 90, 434785)-net in base 64, because

1 times m-reduction [i] would yield (39, 89, 434785)-net in base 64, but
- the generalized Rao bound for nets shows that 64^mÂ â‰¥ 56239 498223 517630 428266 923276 435663 132094 818708 859880 816928 659608 126664 092052 244338 002530 325718 627270 287869 912481 701036 190152 232765 565372 829842 446376 106969 748192 > 64⁸⁹ [i]