MinT - Best Known (138−89, 138, s)-Nets in Base 9

Best Known (138−89, 138, s)-Nets in Base 9

(138−89, 138, 81)-Net over F₉ — Constructive and digital

Digital (49, 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]

(138−89, 138, 168)-Net over F₉ — Digital

Digital (49, 138, 168)-net over F₉, using

net from sequence [i] based on digital (49, 167)-sequence over F₉, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₉ with g(F) = 49 and N(F) ≥ 168, using
  - Drinfeld modules of rank 1 [i]

(138−89, 138, 1991)-Net in Base 9 — Upper bound on s

There is no (49, 138, 1992)-net in base 9, because

1 times m-reduction [i] would yield (49, 137, 1992)-net in base 9, but
- the generalized Rao bound for nets shows that 9^mÂ â‰¥ 54495 646765 933259 152857 144186 690179 362541 682180 053435 836354 925025 388909 789410 007015 337191 728312 774757 409182 706444 296389 491170 149633 > 9¹³⁷ [i]