MinT - Best Known (51, 51+109, s)-Nets in Base 8

Best Known (51, 51+109, s)-Nets in Base 8

(51, 51+109, 98)-Net over F₈ — Constructive and digital

Digital (51, 160, 98)-net over F₈, using

t-expansion [i] based on digital (37, 160, 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+109, 144)-Net over F₈ — Digital

Digital (51, 160, 144)-net over F₈, using

t-expansion [i] based on digital (45, 160, 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+109, 1332)-Net in Base 8 — Upper bound on s

There is no (51, 160, 1333)-net in base 8, because

1 times m-reduction [i] would yield (51, 159, 1333)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 400629 011106 288796 196458 788076 874133 460276 121320 414193 522497 743921 722525 996128 291130 187126 654645 091457 406526 789607 379951 229656 784831 430788 478528 > 8¹⁵⁹ [i]