MinT - Best Known (57, 57+112, s)-Nets in Base 8

Best Known (57, 57+112, s)-Nets in Base 8

(57, 57+112, 98)-Net over F₈ — Constructive and digital

Digital (57, 169, 98)-net over F₈, using

t-expansion [i] based on digital (37, 169, 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+112, 144)-Net over F₈ — Digital

Digital (57, 169, 144)-net over F₈, using

t-expansion [i] based on digital (45, 169, 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+112, 1612)-Net in Base 8 — Upper bound on s

There is no (57, 169, 1613)-net in base 8, because

the generalized Rao bound for nets shows that 8^mÂ â‰¥ 423 851687 754722 013595 787135 740441 454204 580198 238325 818414 538107 477883 115554 521372 190684 026418 381932 053554 118381 040282 262164 815816 230636 724704 449332 430364 > 8¹⁶⁹ [i]