MinT - Best Known (52, 52+105, s)-Nets in Base 8

Best Known (52, 52+105, s)-Nets in Base 8

(52, 52+105, 98)-Net over F₈ — Constructive and digital

Digital (52, 157, 98)-net over F₈, using

t-expansion [i] based on digital (37, 157, 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+105, 144)-Net over F₈ — Digital

Digital (52, 157, 144)-net over F₈, using

t-expansion [i] based on digital (45, 157, 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+105, 1446)-Net in Base 8 — Upper bound on s

There is no (52, 157, 1447)-net in base 8, because

1 times m-reduction [i] would yield (52, 156, 1447)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 771 980505 413412 301422 575671 814574 893238 891044 835537 279623 407998 143116 607402 021156 035820 959901 067284 243112 216507 105845 370295 141375 351145 897512 > 8¹⁵⁶ [i]