MinT - Best Known (41−17, 41, s)-Nets in Base 8

Best Known (41−17, 41, s)-Nets in Base 8

(41−17, 41, 208)-Net over F₈ — Constructive and digital

Digital (24, 41, 208)-net over F₈, using

1 times m-reduction [i] based on digital (24, 42, 208)-net over F₈, using
- trace code for nets [i] based on digital (3, 21, 104)-net over F₆₄, using
  - net from sequence [i] based on digital (3, 103)-sequence over F₆₄, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 3 and N(F) ≥ 104, using
      - a function field by SÃ©mirat [i]

(41−17, 41, 227)-Net over F₈ — Digital

Digital (24, 41, 227)-net over F₈, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(8⁴¹, 227, F₈, 17) (dual of [227, 186, 18]-code), using
- discarding factors / shortening the dual code based on linear OA(8⁴¹, 228, F₈, 17) (dual of [228, 187, 18]-code), using
  - constructionÂ X with VarÅ¡amov bound [i] based on
    1. linear OA(8⁴⁰, 226, F₈, 17) (dual of [226, 186, 18]-code), using
      - trace code [i] based on linear OA(64²⁰, 113, F₆₄, 17) (dual of [113, 93, 18]-code), using
        extended algebraic-geometric code AG_e(F,95P) [i] based on function field F/F₆₄ with g(F) = 3 and N(F) ≥ 113, using
        a curve from the manYPoints database [i]
    2. linear OA(8⁴⁰, 227, F₈, 16) (dual of [227, 187, 17]-code), using Gilbertâ€“VarÅ¡amov bound and b^mÂ = 8⁴⁰Â > V_b^sâˆ’1(kâˆ’1)Â = 467491 200582 635193 749145 627103 353328 [i]
    3. linear OA(8⁰, 1, F₈, 0) (dual of [1, 1, 1]-code), using
      - discarding factors / shortening the dual code based on linear OA(8⁰, s, F₈, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
        OA with only one run / dual of code without redundancy [i]

(41−17, 41, 258)-Net in Base 8 — Constructive

(24, 41, 258)-net in base 8, using

1 times m-reduction [i] based on (24, 42, 258)-net in base 8, using
- trace code for nets [i] based on (3, 21, 129)-net in base 64, using
  - base change [i] based on digital (0, 18, 129)-net over F₁₂₈, using
    - net from sequence [i] based on digital (0, 128)-sequence over F₁₂₈, using
      - generalized Faure sequence [i]
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₂₈ with g(F) = 0 and N(F) ≥ 129, using
        the rational function field F₁₂₈(x) [i]
      - Niederreiter sequence [i]

(41−17, 41, 17616)-Net in Base 8 — Upper bound on s

There is no (24, 41, 17617)-net in base 8, because

1 times m-reduction [i] would yield (24, 40, 17617)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 1 329336 337460 156987 567956 303521 057134 > 8⁴⁰ [i]