MinT - Best Known (57, 57+105, s)-Nets in Base 8

Best Known (57, 57+105, s)-Nets in Base 8

(57, 57+105, 98)-Net over F₈ — Constructive and digital

Digital (57, 162, 98)-net over F₈, using

t-expansion [i] based on digital (37, 162, 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+105, 144)-Net over F₈ — Digital

Digital (57, 162, 144)-net over F₈, using

t-expansion [i] based on digital (45, 162, 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+105, 1774)-Net in Base 8 — Upper bound on s

There is no (57, 162, 1775)-net in base 8, because

1 times m-reduction [i] would yield (57, 161, 1775)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 25 681408 162663 301222 288871 323792 480181 250281 412445 282821 655387 702374 176071 269659 358769 283124 425640 382442 871816 329796 091758 676724 856712 919759 070764 > 8¹⁶¹ [i]