MinT - Best Known (135−83, 135, s)-Nets in Base 9

Best Known (135−83, 135, s)-Nets in Base 9

(135−83, 135, 81)-Net over F₉ — Constructive and digital

Digital (52, 135, 81)-net over F₉, using

t-expansion [i] based on digital (32, 135, 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]

(135−83, 135, 82)-Net in Base 9 — Constructive

(52, 135, 82)-net in base 9, using

base change [i] based on digital (7, 90, 82)-net over F₂₇, using
- net from sequence [i] based on digital (7, 81)-sequence over F₂₇, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₇ with g(F) = 7 and N(F) ≥ 82, using
    - a function field by SÃ©mirat [i]

(135−83, 135, 182)-Net over F₉ — Digital

Digital (52, 135, 182)-net over F₉, using

t-expansion [i] based on digital (50, 135, 182)-net over F₉, using
- net from sequence [i] based on digital (50, 181)-sequence over F₉, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₉ with g(F) = 50 and N(F) ≥ 182, using
    - a curve from the manYPoints database [i]

(135−83, 135, 2626)-Net in Base 9 — Upper bound on s

There is no (52, 135, 2627)-net in base 9, because

1 times m-reduction [i] would yield (52, 134, 2627)-net in base 9, but
- the generalized Rao bound for nets shows that 9^mÂ â‰¥ 73 985521 802521 369773 673991 300096 623852 075206 825304 612768 040415 189714 647655 599094 063898 077886 223604 748099 427051 465485 884207 645657 > 9¹³⁴ [i]