MinT - Best Known (119−59, 119, s)-Nets in Base 3

Best Known (119−59, 119, s)-Nets in Base 3

(119−59, 119, 56)-Net over F₃ — Constructive and digital

Digital (60, 119, 56)-net over F₃, using

1 times m-reduction [i] based on digital (60, 120, 56)-net over F₃, using
- (u,Â u+v)-construction [i] based on
  1. digital (15, 45, 28)-net over F₃, using
    - net from sequence [i] based on digital (15, 27)-sequence over F₃, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 15 and N(F) ≥ 28, using
        an explicitly constructive algebraic function field [i]
  2. digital (15, 75, 28)-net over F₃, using
    - net from sequence [i] based on digital (15, 27)-sequence over F₃ (see above)

(119−59, 119, 67)-Net over F₃ — Digital

Digital (60, 119, 67)-net over F₃, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(3¹¹⁹, 67, F₃, 5, 59) (dual of [(67, 5), 216, 60]-NRT-code), using
- strength reduction [i] based on linear OOA(3¹¹⁹, 67, F₃, 5, 60) (dual of [(67, 5), 216, 61]-NRT-code), using
  - constructionÂ X applied to AG(5;F,254P) âŠ‚ AG(5;F,265P) [i] based on
    1. linear OOA(3¹⁰⁹, 63, F₃, 5, 60) (dual of [(63, 5), 206, 61]-NRT-code), using algebraic-geometric NRT-code AG(5;F,254P) [i] based on function field F/F₃ with g(F) = 49 and N(F) ≥ 64, using
      - a curve from the manYPoints database [i]
    2. linear OOA(3⁹⁸, 63, F₃, 5, 49) (dual of [(63, 5), 217, 50]-NRT-code), using algebraic-geometric NRT-code AG(5;F,265P) [i] based on function field F/F₃ with g(F) = 49 and N(F) ≥ 64 (see above)
    3. linear OOA(3¹⁰, 4, F₃, 5, 10) (dual of [(4, 5), 10, 11]-NRT-code), using
      - extended Reedâ€“Solomon NRT-code RS_e(5;10,3) [i]

(119−59, 119, 482)-Net in Base 3 — Upper bound on s

There is no (60, 119, 483)-net in base 3, because

1 times m-reduction [i] would yield (60, 118, 483)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 210 849157 003310 518448 914267 907762 291002 587705 849300 078799 > 3¹¹⁸ [i]