MinT - Best Known (82−45, 82, s)-Nets in Base 64

Best Known (82−45, 82, s)-Nets in Base 64

(82−45, 82, 513)-Net over F₆₄ — Constructive and digital

Digital (37, 82, 513)-net over F₆₄, using

t-expansion [i] based on digital (28, 82, 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]

(82−45, 82, 658)-Net over F₆₄ — Digital

Digital (37, 82, 658)-net over F₆₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(82−45, 82, 641994)-Net in Base 64 — Upper bound on s

There is no (37, 82, 641995)-net in base 64, because

1 times m-reduction [i] would yield (37, 81, 641995)-net in base 64, but
- the generalized Rao bound for nets shows that 64^mÂ â‰¥ 199 794331 377690 808294 504469 200183 623803 819514 697376 981984 832049 046965 716707 446657 474895 381234 816142 444409 662160 914161 475821 170497 724624 775795 177224 > 64⁸¹ [i]