MinT - Best Known (138−98, 138, s)-Nets in Base 9

Best Known (138−98, 138, s)-Nets in Base 9

(138−98, 138, 81)-Net over F₉ — Constructive and digital

Digital (40, 138, 81)-net over F₉, using

t-expansion [i] based on digital (32, 138, 81)-net over F₉, using
- net from sequence [i] based on digital (32, 80)-sequence over F₉, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₉ with g(F) = 32 and N(F) ≥ 81, using
    - F₄ from the tower of function fields by Bezerra and GarcÃa over F₉ [i]

(138−98, 138, 140)-Net over F₉ — Digital

Digital (40, 138, 140)-net over F₉, using

t-expansion [i] based on digital (39, 138, 140)-net over F₉, using
- net from sequence [i] based on digital (39, 139)-sequence over F₉, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₉ with g(F) = 39 and N(F) ≥ 140, using
    - a curve from the manYPoints database [i]

(138−98, 138, 1133)-Net in Base 9 — Upper bound on s

There is no (40, 138, 1134)-net in base 9, because

the generalized Rao bound for nets shows that 9^mÂ â‰¥ 497920 513474 825348 255485 387729 715058 010635 887051 735040 870699 159655 527216 298823 110578 832139 068651 588051 700614 094239 699936 054906 584689 > 9¹³⁸ [i]