MinT - Best Known (138−95, 138, s)-Nets in Base 9

Best Known (138−95, 138, s)-Nets in Base 9

(138−95, 138, 81)-Net over F₉ — Constructive and digital

Digital (43, 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]

(138−95, 138, 147)-Net over F₉ — Digital

Digital (43, 138, 147)-net over F₉, using

net from sequence [i] based on digital (43, 146)-sequence over F₉, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₉ with g(F) = 43 and N(F) ≥ 147, using
  - a curve from the manYPoints database [i]

(138−95, 138, 1359)-Net in Base 9 — Upper bound on s

There is no (43, 138, 1360)-net in base 9, because

1 times m-reduction [i] would yield (43, 137, 1360)-net in base 9, but
- the generalized Rao bound for nets shows that 9^mÂ â‰¥ 54346 192979 605710 274701 664146 358989 608867 373955 237999 208583 807516 754405 343879 828482 303132 384050 443743 280362 276446 616799 402959 906689 > 9¹³⁷ [i]