MinT - Best Known (156−101, 156, s)-Nets in Base 8

Best Known (156−101, 156, s)-Nets in Base 8

(156−101, 156, 98)-Net over F₈ — Constructive and digital

Digital (55, 156, 98)-net over F₈, using

t-expansion [i] based on digital (37, 156, 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]

(156−101, 156, 144)-Net over F₈ — Digital

Digital (55, 156, 144)-net over F₈, using

t-expansion [i] based on digital (45, 156, 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]

(156−101, 156, 1722)-Net in Base 8 — Upper bound on s

There is no (55, 156, 1723)-net in base 8, because

1 times m-reduction [i] would yield (55, 155, 1723)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 95 309177 809260 767732 444475 274407 108497 043518 973124 916349 993418 082463 942053 047986 993082 803417 722937 438264 266374 209433 025062 553530 341561 771627 > 8¹⁵⁵ [i]