MinT - Best Known (18, 18+11, s)-Nets in Base 4

Best Known (18, 18+11, s)-Nets in Base 4

(18, 18+11, 76)-Net over F₄ — Constructive and digital

Digital (18, 29, 76)-net over F₄, using

1 times m-reduction [i] based on digital (18, 30, 76)-net over F₄, using
- trace code for nets [i] based on digital (3, 15, 38)-net over F₁₆, using
  - net from sequence [i] based on digital (3, 37)-sequence over F₁₆, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₆ with g(F) = 3 and N(F) ≥ 38, using
      - a function field by SÃ©mirat [i]

(18, 18+11, 98)-Net over F₄ — Digital

Digital (18, 29, 98)-net over F₄, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(4²⁹, 98, F₄, 11) (dual of [98, 69, 12]-code), using
- discarding factors / shortening the dual code based on linear OA(4²⁹, 104, F₄, 11) (dual of [104, 75, 12]-code), using
  - constructionÂ X with VarÅ¡amov bound [i] based on
    1. linear OA(4²⁸, 102, F₄, 11) (dual of [102, 74, 12]-code), using
      - a â€œTolâ€ code from Brouwerâ€™s database [i]
    2. linear OA(4²⁸, 103, F₄, 10) (dual of [103, 75, 11]-code), using Gilbertâ€“VarÅ¡amov bound and b^mÂ = 4²⁸Â > V_b^sâˆ’1(kâˆ’1)Â = 46562 425943 317876 [i]
    3. linear OA(4⁰, 1, F₄, 0) (dual of [1, 1, 1]-code), using
      - discarding factors / shortening the dual code based on linear OA(4⁰, s, F₄, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
        OA with only one run / dual of code without redundancy [i]

(18, 18+11, 2039)-Net in Base 4 — Upper bound on s

There is no (18, 29, 2040)-net in base 4, because

1 times m-reduction [i] would yield (18, 28, 2040)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 72189 246012 254695 > 4²⁸ [i]