MinT - Best Known (24, 24+17, s)-Nets in Base 9

Best Known (24, 24+17, s)-Nets in Base 9

(24, 24+17, 232)-Net over F₉ — Constructive and digital

Digital (24, 41, 232)-net over F₉, using

3 times m-reduction [i] based on digital (24, 44, 232)-net over F₉, using
- trace code for nets [i] based on digital (2, 22, 116)-net over F₈₁, using
  - net from sequence [i] based on digital (2, 115)-sequence over F₈₁, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈₁ with g(F) = 2 and N(F) ≥ 116, using
      - a function field by SÃ©mirat [i]

(24, 24+17, 274)-Net over F₉ — Digital

Digital (24, 41, 274)-net over F₉, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(9⁴¹, 274, F₉, 17) (dual of [274, 233, 18]-code), using
- constructionÂ X with VarÅ¡amov bound [i] based on
  1. linear OA(9⁴⁰, 272, F₉, 17) (dual of [272, 232, 18]-code), using
    - trace code [i] based on linear OA(81²⁰, 136, F₈₁, 17) (dual of [136, 116, 18]-code), using
      - extended algebraic-geometric code AG_e(F,118P) [i] based on function field F/F₈₁ with g(F) = 3 and N(F) ≥ 136, using
        a curve from the manYPoints database [i]
  2. linear OA(9⁴⁰, 273, F₉, 16) (dual of [273, 233, 17]-code), using Gilbertâ€“VarÅ¡amov bound and b^mÂ = 9⁴⁰Â > V_b^sâˆ’1(kâˆ’1)Â = 60 372393 163275 215796 293607 929580 627585 [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]

(24, 24+17, 27780)-Net in Base 9 — Upper bound on s

There is no (24, 41, 27781)-net in base 9, because

1 times m-reduction [i] would yield (24, 40, 27781)-net in base 9, but
- the generalized Rao bound for nets shows that 9^mÂ â‰¥ 147 824042 125039 000536 586764 991565 552193 > 9⁴⁰ [i]