MinT - Best Known (15, 15+10, s)-Nets in Base 7

Best Known (15, 15+10, s)-Nets in Base 7

(15, 15+10, 108)-Net over F₇ — Constructive and digital

Digital (15, 25, 108)-net over F₇, using

(u,Â u+v)-construction [i] based on
1. digital (0, 5, 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 (10, 20, 100)-net over F₇, using
  - trace code for nets [i] based on digital (0, 10, 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]

(15, 15+10, 211)-Net over F₇ — Digital

Digital (15, 25, 211)-net over F₇, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(7²⁵, 211, F₇, 10) (dual of [211, 186, 11]-code), using
- discarding factors / shortening the dual code based on linear OA(7²⁵, 342, F₇, 10) (dual of [342, 317, 11]-code), using
  - the primitive expurgated narrow-sense BCH-code C(I) with length 342Â = 7³âˆ’1, defining interval IÂ =Â [0,9], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 11 [i]

(15, 15+10, 7294)-Net in Base 7 — Upper bound on s

There is no (15, 25, 7295)-net in base 7, because

the generalized Rao bound for nets shows that 7^mÂ â‰¥ 1341 357648 037214 494435 > 7²⁵ [i]