MinT - Best Known (51, 51+113, s)-Nets in Base 8

Best Known (51, 51+113, s)-Nets in Base 8

(51, 51+113, 98)-Net over F₈ — Constructive and digital

Digital (51, 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]

(51, 51+113, 144)-Net over F₈ — Digital

Digital (51, 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]

(51, 51+113, 1283)-Net in Base 8 — Upper bound on s

There is no (51, 164, 1284)-net in base 8, because

1 times m-reduction [i] would yield (51, 163, 1284)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 1626 556963 218401 129002 522039 284078 791466 550490 895692 011686 894100 036334 662045 247597 595069 010205 733778 972044 316139 895402 926556 401692 899215 479788 597280 > 8¹⁶³ [i]