MinT - Best Known (10, 10+7, s)-Nets in Base 7

Best Known (10, 10+7, s)-Nets in Base 7

(10, 10+7, 108)-Net over F₇ — Constructive and digital

Digital (10, 17, 108)-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 (7, 14, 100)-net over F₇, using
  - trace code for nets [i] based on digital (0, 7, 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]

(10, 10+7, 170)-Net over F₇ — Digital

Digital (10, 17, 170)-net over F₇, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(7¹⁷, 170, F₇, 7) (dual of [170, 153, 8]-code), using
- 1 times truncation [i] based on linear OA(7¹⁸, 171, F₇, 8) (dual of [171, 153, 9]-code), using
  - the BCH-code C(I) with length 171Â | 7³âˆ’1, defining interval IÂ =Â {1,17,33,…,113}, and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 9 [i]

(10, 10+7, 9735)-Net in Base 7 — Upper bound on s

There is no (10, 17, 9736)-net in base 7, because

1 times m-reduction [i] would yield (10, 16, 9736)-net in base 7, but
- the generalized Rao bound for nets shows that 7^mÂ â‰¥ 33 238766 621953 > 7¹⁶ [i]