MinT - Best Known (25, 25+15, s)-Nets in Base 7

Best Known (25, 25+15, s)-Nets in Base 7

(25, 25+15, 116)-Net over F₇ — Constructive and digital

Digital (25, 40, 116)-net over F₇, using

(u,Â u+v)-construction [i] based on
1. digital (3, 10, 16)-net over F₇, using
  - (u,Â u+v)-construction [i] based on
    1. digital (0, 3, 8)-net over F₇, using
      - net from sequence [i] based on digital (0, 7)-sequence over F₇, using
        Faure sequence [i]
        Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₇ with g(F) = 0 and N(F) ≥ 8, using
        the rational function field F₇(x) [i]
        Niederreiter sequence [i]
    2. digital (0, 7, 8)-net over F₇, using
      - net from sequence [i] based on digital (0, 7)-sequence over F₇ (see above)
2. digital (15, 30, 100)-net over F₇, using
  - trace code for nets [i] based on digital (0, 15, 50)-net over F₄₉, using
    - net from sequence [i] based on digital (0, 49)-sequence over F₄₉, using
      - generalized Faure sequence [i]
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄₉ with g(F) = 0 and N(F) ≥ 50, using
        the rational function field F₄₉(x) [i]
      - Niederreiter sequence [i]

(25, 25+15, 317)-Net over F₇ — Digital

Digital (25, 40, 317)-net over F₇, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(7⁴⁰, 317, F₇, 15) (dual of [317, 277, 16]-code), using
- discarding factors / shortening the dual code based on linear OA(7⁴⁰, 342, F₇, 15) (dual of [342, 302, 16]-code), using
  - the primitive BCH-code C(I) with length 342Â = 7³âˆ’1, defining interval IÂ =Â {−3,−2,…,11}, and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 16 [i]

(25, 25+15, 28781)-Net in Base 7 — Upper bound on s

There is no (25, 40, 28782)-net in base 7, because

1 times m-reduction [i] would yield (25, 39, 28782)-net in base 7, but
- the generalized Rao bound for nets shows that 7^mÂ â‰¥ 909 735291 674457 068393 375804 490529 > 7³⁹ [i]