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

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

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

Digital (59, 118, 56)-net over F₃, using

(u,Â u+v)-construction [i] based on
1. digital (15, 44, 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, 74, 28)-net over F₃, using
  - net from sequence [i] based on digital (15, 27)-sequence over F₃ (see above)

(59, 59+59, 65)-Net over F₃ — Digital

Digital (59, 118, 65)-net over F₃, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(3¹¹⁸, 65, F₃, 5, 59) (dual of [(65, 5), 207, 60]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(3¹¹⁸, 67, F₃, 5, 59) (dual of [(67, 5), 217, 60]-NRT-code), using
  - constructionÂ X applied to AG(5;F,255P) âŠ‚ AG(5;F,266P) [i] based on
    1. linear OOA(3¹⁰⁸, 63, F₃, 5, 59) (dual of [(63, 5), 207, 60]-NRT-code), using algebraic-geometric NRT-code AG(5;F,255P) [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(3⁹⁷, 63, F₃, 5, 48) (dual of [(63, 5), 218, 49]-NRT-code), using algebraic-geometric NRT-code AG(5;F,266P) [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]

(59, 59+59, 463)-Net in Base 3 — Upper bound on s

There is no (59, 118, 464)-net in base 3, because

1 times m-reduction [i] would yield (59, 117, 464)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 70 145043 554945 794520 040194 956781 859287 810867 830014 501537 > 3¹¹⁷ [i]