MinT - Best Known (57, 57+107, s)-Nets in Base 8

Best Known (57, 57+107, s)-Nets in Base 8

(57, 57+107, 98)-Net over F₈ — Constructive and digital

Digital (57, 164, 98)-net over F₈, using

t-expansion [i] based on digital (37, 164, 98)-net over F₈, using
- net from sequence [i] based on digital (37, 97)-sequence over F₈, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈ with g(F) = 37 and N(F) ≥ 98, using
    - F₅ from the tower of function fields over F₈ by van der Geer and van der Vlugt [i]

(57, 57+107, 144)-Net over F₈ — Digital

Digital (57, 164, 144)-net over F₈, using

t-expansion [i] based on digital (45, 164, 144)-net over F₈, using
- net from sequence [i] based on digital (45, 143)-sequence over F₈, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈ with g(F) = 45 and N(F) ≥ 144, using
    - Drinfeld modules of rank 1 [i]

(57, 57+107, 1729)-Net in Base 8 — Upper bound on s

There is no (57, 164, 1730)-net in base 8, because

1 times m-reduction [i] would yield (57, 163, 1730)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 1642 830355 706606 264252 347570 916411 372192 193101 881360 358112 704877 203646 242642 588591 090056 901285 179501 304850 374465 580847 662649 520282 971052 189202 064656 > 8¹⁶³ [i]