MinT - Best Known (28, 28+12, s)-Nets in Base 7

Best Known (28, 28+12, s)-Nets in Base 7

(28, 28+12, 208)-Net over F₇ — Constructive and digital

Digital (28, 40, 208)-net over F₇, using

generalized (u,Â u+v)-construction [i] based on
1. digital (0, 4, 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 (6, 12, 100)-net over F₇, using
  - trace code for nets [i] based on digital (0, 6, 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]
3. digital (12, 24, 100)-net over F₇, using
  - trace code for nets [i] based on digital (0, 12, 50)-net over F₄₉, using
    - net from sequence [i] based on digital (0, 49)-sequence over F₄₉ (see above)

(28, 28+12, 1201)-Net over F₇ — Digital

Digital (28, 40, 1201)-net over F₇, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(7⁴⁰, 1201, F₇, 12) (dual of [1201, 1161, 13]-code), using
- the BCH-code C(I) with length 1201Â | 7⁸âˆ’1, defining interval IÂ =Â {1,110,219,…,1200}, and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 13 [i]

(28, 28+12, 214808)-Net in Base 7 — Upper bound on s

There is no (28, 40, 214809)-net in base 7, because

the generalized Rao bound for nets shows that 7^mÂ â‰¥ 6366 962654 497181 862399 747500 000065 > 7⁴⁰ [i]