MinT - Best Known (56, 56+62, s)-Nets in Base 9

Best Known (56, 56+62, s)-Nets in Base 9

(56, 56+62, 114)-Net over F₉ — Constructive and digital

Digital (56, 118, 114)-net over F₉, using

(u,Â u+v)-construction [i] based on
1. digital (8, 39, 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, 79, 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]

(56, 56+62, 185)-Net over F₉ — Digital

Digital (56, 118, 185)-net over F₉, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(9¹¹⁸, 185, F₉, 3, 62) (dual of [(185, 3), 437, 63]-NRT-code), using
- constructionÂ X applied to AG(3;F,480P) âŠ‚ AG(3;F,487P) [i] based on
  1. linear OOA(9¹¹², 181, F₉, 3, 62) (dual of [(181, 3), 431, 63]-NRT-code), using algebraic-geometric NRT-code AG(3;F,480P) [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(9¹⁰⁵, 181, F₉, 3, 55) (dual of [(181, 3), 438, 56]-NRT-code), using algebraic-geometric NRT-code AG(3;F,487P) [i] based on function field F/F₉ with g(F) = 50 and N(F) ≥ 182 (see above)
  3. linear OOA(9⁶, 4, F₉, 3, 6) (dual of [(4, 3), 6, 7]-NRT-code), using
    - discarding factors / shortening the dual code based on linear OOA(9⁶, 9, F₉, 3, 6) (dual of [(9, 3), 21, 7]-NRT-code), using
      - Reedâ€“Solomon NRT-code RS(3;21,9) [i]

(56, 56+62, 6637)-Net in Base 9 — Upper bound on s

There is no (56, 118, 6638)-net in base 9, because

the generalized Rao bound for nets shows that 9^mÂ â‰¥ 40005 371505 398184 390224 473870 078711 060400 128182 936237 827450 350853 088548 810015 220649 600957 321895 984734 030325 774545 > 9¹¹⁸ [i]