MinT - Best Known (85−45, 85, s)-Nets in Base 64

Best Known (85−45, 85, s)-Nets in Base 64

(85−45, 85, 513)-Net over F₆₄ — Constructive and digital

Digital (40, 85, 513)-net over F₆₄, using

t-expansion [i] based on digital (28, 85, 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]

(85−45, 85, 867)-Net over F₆₄ — Digital

Digital (40, 85, 867)-net over F₆₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(85−45, 85, 1131962)-Net in Base 64 — Upper bound on s

There is no (40, 85, 1131963)-net in base 64, because

1 times m-reduction [i] would yield (40, 84, 1131963)-net in base 64, but
- the generalized Rao bound for nets shows that 64^mÂ â‰¥ 52 375256 353896 990968 507294 164969 887947 081161 112015 436177 060417 674220 112488 614209 118704 685954 441843 915094 663162 815769 946787 873804 316051 810746 776045 151938 > 64⁸⁴ [i]