MinT - Best Known (89−57, 89, s)-Nets in Base 64

Best Known (89−57, 89, s)-Nets in Base 64

(89−57, 89, 513)-Net over F₆₄ — Constructive and digital

Digital (32, 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]

(89−57, 89, 85144)-Net in Base 64 — Upper bound on s

There is no (32, 89, 85145)-net in base 64, because

1 times m-reduction [i] would yield (32, 88, 85145)-net in base 64, but
- the generalized Rao bound for nets shows that 64^mÂ â‰¥ 878 702876 551373 264282 427048 238482 765675 031462 485647 276526 598718 003744 096245 224689 782034 200451 269168 303595 458585 097302 983188 012478 255566 590587 587899 760011 686304 > 64⁸⁸ [i]