MinT - Best Known (89−49, 89, s)-Nets in Base 64

Best Known (89−49, 89, s)-Nets in Base 64

(89−49, 89, 513)-Net over F₆₄ — Constructive and digital

Digital (40, 89, 513)-net over F₆₄, using

t-expansion [i] based on digital (28, 89, 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]

(89−49, 89, 682)-Net over F₆₄ — Digital

Digital (40, 89, 682)-net over F₆₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(89−49, 89, 652640)-Net in Base 64 — Upper bound on s

There is no (40, 89, 652641)-net in base 64, because

1 times m-reduction [i] would yield (40, 88, 652641)-net in base 64, but
- the generalized Rao bound for nets shows that 64^mÂ â‰¥ 878 702257 553392 535694 457117 404214 118116 339020 079959 478118 871022 020431 542343 161750 051166 394284 648439 098193 350619 582516 683944 716465 947554 629996 971623 639715 134564 > 64⁸⁸ [i]