MinT - Best Known (54, 54+59, s)-Nets in Base 9

Best Known (54, 54+59, s)-Nets in Base 9

(54, 54+59, 114)-Net over F₉ — Constructive and digital

Digital (54, 113, 114)-net over F₉, using

(u,Â u+v)-construction [i] based on
1. digital (8, 37, 40)-net over F₉, using
  - net from sequence [i] based on digital (8, 39)-sequence over F₉, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₉ with g(F) = 8 and N(F) ≥ 40, using
      - a function field by SÃ©mirat [i]
2. digital (17, 76, 74)-net over F₉, using
  - net from sequence [i] based on digital (17, 73)-sequence over F₉, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₉ with g(F) = 17 and N(F) ≥ 74, using
      - a function field by GarcÃa and Quoos [i]

(54, 54+59, 184)-Net over F₉ — Digital

Digital (54, 113, 184)-net over F₉, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(9¹¹³, 184, F₉, 3, 59) (dual of [(184, 3), 439, 60]-NRT-code), using
- constructionÂ X applied to AG(3;F,483P) âŠ‚ AG(3;F,488P) [i] based on
  1. linear OOA(9¹⁰⁹, 181, F₉, 3, 59) (dual of [(181, 3), 434, 60]-NRT-code), using algebraic-geometric NRT-code AG(3;F,483P) [i] based on function field F/F₉ with g(F) = 50 and N(F) ≥ 182, using
    - a curve from the manYPoints database [i]
  2. linear OOA(9¹⁰⁴, 181, F₉, 3, 54) (dual of [(181, 3), 439, 55]-NRT-code), using algebraic-geometric NRT-code AG(3;F,488P) [i] based on function field F/F₉ with g(F) = 50 and N(F) ≥ 182 (see above)
  3. linear OOA(9⁴, 3, F₉, 3, 4) (dual of [(3, 3), 5, 5]-NRT-code), using
    - discarding factors / shortening the dual code based on linear OOA(9⁴, 9, F₉, 3, 4) (dual of [(9, 3), 23, 5]-NRT-code), using
      - Reedâ€“Solomon NRT-code RS(3;23,9) [i]

(54, 54+59, 7051)-Net in Base 9 — Upper bound on s

There is no (54, 113, 7052)-net in base 9, because

1 times m-reduction [i] would yield (54, 112, 7052)-net in base 9, but
- the generalized Rao bound for nets shows that 9^mÂ â‰¥ 75083 165381 665427 399643 052798 449093 872577 460665 855862 731518 385699 054509 366509 638797 624269 018015 276265 030625 > 9¹¹² [i]