MinT - Best Known (69−40, 69, s)-Nets in Base 81

Best Known (69−40, 69, s)-Nets in Base 81

(69−40, 69, 370)-Net over F₈₁ — Constructive and digital

Digital (29, 69, 370)-net over F₈₁, using

t-expansion [i] based on digital (16, 69, 370)-net over F₈₁, using
- net from sequence [i] based on digital (16, 369)-sequence over F₈₁, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈₁ with g(F) = 16 and N(F) ≥ 370, using
    - a function field by GarcÃa and Quoos [i]

(69−40, 69, 502)-Net over F₈₁ — Digital

Digital (29, 69, 502)-net over F₈₁, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(81⁶⁹, 502, F₈₁, 2, 40) (dual of [(502, 2), 935, 41]-NRT-code), using
- constructionÂ X applied to AG(2;F,957P) âŠ‚ AG(2;F,961P) [i] based on
  1. linear OOA(81⁶⁶, 499, F₈₁, 2, 40) (dual of [(499, 2), 932, 41]-NRT-code), using algebraic-geometric NRT-code AG(2;F,957P) [i] based on function field F/F₈₁ with g(F) = 26 and N(F) ≥ 500, using
    - a curve from the manYPoints database [i]
  2. linear OOA(81⁶², 499, F₈₁, 2, 36) (dual of [(499, 2), 936, 37]-NRT-code), using algebraic-geometric NRT-code AG(2;F,961P) [i] based on function field F/F₈₁ with g(F) = 26 and N(F) ≥ 500 (see above)
  3. linear OOA(81³, 3, F₈₁, 2, 3) (dual of [(3, 2), 3, 4]-NRT-code), using
    - discarding factors / shortening the dual code based on linear OOA(81³, 81, F₈₁, 2, 3) (dual of [(81, 2), 159, 4]-NRT-code), using
      - Reedâ€“Solomon NRT-code RS(2;159,81) [i]

(69−40, 69, 398546)-Net in Base 81 — Upper bound on s

There is no (29, 69, 398547)-net in base 81, because

the generalized Rao bound for nets shows that 81^mÂ â‰¥ 484706 113747 021993 934441 285354 313960 055259 706357 607572 369078 827338 281473 970700 050857 012410 651258 720418 234764 153995 752608 709601 643201 > 81⁶⁹ [i]