MinT - Best Known (52, 52+113, s)-Nets in Base 8

Best Known (52, 52+113, s)-Nets in Base 8

(52, 52+113, 98)-Net over F₈ — Constructive and digital

Digital (52, 165, 98)-net over F₈, using

t-expansion [i] based on digital (37, 165, 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]

(52, 52+113, 144)-Net over F₈ — Digital

Digital (52, 165, 144)-net over F₈, using

t-expansion [i] based on digital (45, 165, 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]

(52, 52+113, 1333)-Net in Base 8 — Upper bound on s

There is no (52, 165, 1334)-net in base 8, because

1 times m-reduction [i] would yield (52, 164, 1334)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 13067 536883 110751 388476 922192 517508 245524 833305 635934 506182 337448 950222 219741 664254 370463 351503 324507 344680 916013 946972 272960 093866 329462 451267 816160 > 8¹⁶⁴ [i]