MinT - Best Known (25−10, 25, s)-Nets in Base 4

Best Known (25−10, 25, s)-Nets in Base 4

(25−10, 25, 66)-Net over F₄ — Constructive and digital

Digital (15, 25, 66)-net over F₄, using

1 times m-reduction [i] based on digital (15, 26, 66)-net over F₄, using
- trace code for nets [i] based on digital (2, 13, 33)-net over F₁₆, using
  - net from sequence [i] based on digital (2, 32)-sequence over F₁₆, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₆ with g(F) = 2 and N(F) ≥ 33, using
      - a function field by SÃ©mirat [i]

(25−10, 25, 76)-Net over F₄ — Digital

Digital (15, 25, 76)-net over F₄, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(4²⁵, 76, F₄, 10) (dual of [76, 51, 11]-code), using
- constructionÂ X with VarÅ¡amov bound [i] based on
  1. linear OA(4²⁴, 74, F₄, 10) (dual of [74, 50, 11]-code), using
    - BCH codes lengthened with UEP codes [i]
  2. linear OA(4²⁴, 75, F₄, 9) (dual of [75, 51, 10]-code), using Gilbertâ€“VarÅ¡amov bound and b^mÂ = 4²⁴Â > V_b^sâˆ’1(kâˆ’1)Â = 102 958681 891921 [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]

(25−10, 25, 885)-Net in Base 4 — Upper bound on s

There is no (15, 25, 886)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 1128 617315 097235 > 4²⁵ [i]