MinT - Best Known (78−41, 78, s)-Nets in Base 64

Best Known (78−41, 78, s)-Nets in Base 64

(78−41, 78, 513)-Net over F₆₄ — Constructive and digital

Digital (37, 78, 513)-net over F₆₄, using

t-expansion [i] based on digital (28, 78, 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]

(78−41, 78, 853)-Net over F₆₄ — Digital

Digital (37, 78, 853)-net over F₆₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(78−41, 78, 1185099)-Net in Base 64 — Upper bound on s

There is no (37, 78, 1185100)-net in base 64, because

1 times m-reduction [i] would yield (37, 77, 1185100)-net in base 64, but
- the generalized Rao bound for nets shows that 64^mÂ â‰¥ 11 908533 667319 571448 921263 729821 346190 143583 465812 762549 217669 941052 814500 022739 453096 282995 042648 461781 152284 460591 935332 309573 932784 801656 > 64⁷⁷ [i]