MinT - Best Known (31, 78, s)-Nets in Base 64

Best Known (31, 78, s)-Nets in Base 64

(31, 78, 513)-Net over F₆₄ — Constructive and digital

Digital (31, 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]

(31, 78, 166671)-Net in Base 64 — Upper bound on s

There is no (31, 78, 166672)-net in base 64, because

1 times m-reduction [i] would yield (31, 77, 166672)-net in base 64, but
- the generalized Rao bound for nets shows that 64^mÂ â‰¥ 11 908585 266077 455140 813484 043595 954357 709099 190541 547839 009199 405794 202426 807545 824393 998646 144438 894615 835050 607413 146159 155566 047624 711408 > 64⁷⁷ [i]