MinT - Best Known (34, 75, s)-Nets in Base 8

Best Known (34, 75, s)-Nets in Base 8

(34, 75, 74)-Net over F₈ — Constructive and digital

Digital (34, 75, 74)-net over F₈, using

(u,Â u+v)-construction [i] based on
1. digital (0, 20, 9)-net over F₈, using
  - net from sequence [i] based on digital (0, 8)-sequence over F₈, using
    - generalized Faure sequence [i]
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈ with g(F) = 0 and N(F) ≥ 9, using
      - the rational function field F₈(x) [i]
    - Niederreiter sequence [i]
2. digital (14, 55, 65)-net over F₈, using
  - net from sequence [i] based on digital (14, 64)-sequence over F₈, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈ with g(F) = 14 and N(F) ≥ 65, using
      - the Suzuki function field over F₈ [i]

(34, 75, 99)-Net over F₈ — Digital

Digital (34, 75, 99)-net over F₈, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(8⁷⁵, 99, F₈, 3, 41) (dual of [(99, 3), 222, 42]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(8⁷⁵, 100, F₈, 3, 41) (dual of [(100, 3), 225, 42]-NRT-code), using
  - constructionÂ X applied to AG(3;F,243P) âŠ‚ AG(3;F,251P) [i] based on
    1. linear OOA(8⁶⁸, 95, F₈, 3, 41) (dual of [(95, 3), 217, 42]-NRT-code), using algebraic-geometric NRT-code AG(3;F,243P) [i] based on function field F/F₈ with g(F) = 27 and N(F) ≥ 96, using
      - a curve from the manYPoints database [i]
    2. linear OOA(8⁶⁰, 95, F₈, 3, 33) (dual of [(95, 3), 225, 34]-NRT-code), using algebraic-geometric NRT-code AG(3;F,251P) [i] based on function field F/F₈ with g(F) = 27 and N(F) ≥ 96 (see above)
    3. linear OOA(8⁷, 5, F₈, 3, 7) (dual of [(5, 3), 8, 8]-NRT-code), using
      - discarding factors / shortening the dual code based on linear OOA(8⁷, 8, F₈, 3, 7) (dual of [(8, 3), 17, 8]-NRT-code), using
        Reedâ€“Solomon NRT-code RS(3;17,8) [i]

(34, 75, 2591)-Net in Base 8 — Upper bound on s

There is no (34, 75, 2592)-net in base 8, because

1 times m-reduction [i] would yield (34, 74, 2592)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 6 757500 484165 917174 553791 428350 926718 930480 560122 794737 048146 570203 > 8⁷⁴ [i]