MinT - Best Known (114−67, 114, s)-Nets in Base 9

Best Known (114−67, 114, s)-Nets in Base 9

(114−67, 114, 81)-Net over F₉ — Constructive and digital

Digital (47, 114, 81)-net over F₉, using

t-expansion [i] based on digital (32, 114, 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]

(114−67, 114, 88)-Net in Base 9 — Constructive

(47, 114, 88)-net in base 9, using

base change [i] based on digital (9, 76, 88)-net over F₂₇, using
- net from sequence [i] based on digital (9, 87)-sequence over F₂₇, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₇ with g(F) = 9 and N(F) ≥ 88, using
    - a function field by SÃ©mirat [i]

(114−67, 114, 162)-Net over F₉ — Digital

Digital (47, 114, 162)-net over F₉, using

t-expansion [i] based on digital (46, 114, 162)-net over F₉, using
- net from sequence [i] based on digital (46, 161)-sequence over F₉, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₉ with g(F) = 46 and N(F) ≥ 162, using
    - a curve from the manYPoints database [i]

(114−67, 114, 3026)-Net in Base 9 — Upper bound on s

There is no (47, 114, 3027)-net in base 9, because

1 times m-reduction [i] would yield (47, 113, 3027)-net in base 9, but
- the generalized Rao bound for nets shows that 9^mÂ â‰¥ 678388 852355 772400 818423 685117 486386 591739 459347 014204 682993 039978 442045 334051 048313 431344 031315 575607 552409 > 9¹¹³ [i]