MinT - Best Known (63, 63+109, s)-Nets in Base 8

Best Known (63, 63+109, s)-Nets in Base 8

(63, 63+109, 98)-Net over F₈ — Constructive and digital

Digital (63, 172, 98)-net over F₈, using

t-expansion [i] based on digital (37, 172, 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]

(63, 63+109, 144)-Net over F₈ — Digital

Digital (63, 172, 144)-net over F₈, using

t-expansion [i] based on digital (45, 172, 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]

(63, 63+109, 2134)-Net in Base 8 — Upper bound on s

There is no (63, 172, 2135)-net in base 8, because

1 times m-reduction [i] would yield (63, 171, 2135)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 26949 493199 385708 873757 986156 600344 995589 948150 131418 676665 079436 821470 413558 738208 401999 811032 028636 465513 823066 179084 184250 930509 239352 210718 019670 495554 > 8¹⁷¹ [i]