MinT - Best Known (44, 75, s)-Nets in Base 9

Best Known (44, 75, s)-Nets in Base 9

(44, 75, 320)-Net over F₉ — Constructive and digital

Digital (44, 75, 320)-net over F₉, using

3 times m-reduction [i] based on digital (44, 78, 320)-net over F₉, using
- trace code for nets [i] based on digital (5, 39, 160)-net over F₈₁, using
  - net from sequence [i] based on digital (5, 159)-sequence over F₈₁, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈₁ with g(F) = 5 and N(F) ≥ 160, using
      - a function field by SÃ©mirat [i]

(44, 75, 382)-Net over F₉ — Digital

Digital (44, 75, 382)-net over F₉, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(9⁷⁵, 382, F₉, 31) (dual of [382, 307, 32]-code), using
- constructionÂ X with VarÅ¡amov bound [i] based on
  1. linear OA(9⁷⁴, 380, F₉, 31) (dual of [380, 306, 32]-code), using
    - trace code [i] based on linear OA(81³⁷, 190, F₈₁, 31) (dual of [190, 153, 32]-code), using
      - extended algebraic-geometric code AG_e(F,158P) [i] based on function field F/F₈₁ with g(F) = 6 and N(F) ≥ 190, using
        a curve from the manYPoints database [i]
  2. linear OA(9⁷⁴, 381, F₉, 30) (dual of [381, 307, 31]-code), using Gilbertâ€“VarÅ¡amov bound and b^mÂ = 9⁷⁴Â > V_b^sâˆ’1(kâˆ’1)Â = 3847 901680 424709 835126 936895 014502 816690 241341 511126 673552 677275 919969 [i]
  3. linear OA(9⁰, 1, F₉, 0) (dual of [1, 1, 1]-code), using
    - discarding factors / shortening the dual code based on linear OA(9⁰, s, F₉, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
      - OA with only one run / dual of code without redundancy [i]

(44, 75, 40942)-Net in Base 9 — Upper bound on s

There is no (44, 75, 40943)-net in base 9, because

1 times m-reduction [i] would yield (44, 74, 40943)-net in base 9, but
- the generalized Rao bound for nets shows that 9^mÂ â‰¥ 41110 898058 168720 338490 549046 699339 395213 211663 684951 867546 061962 050057 > 9⁷⁴ [i]