MinT - Best Known (31, 31+20, s)-Nets in Base 7

Best Known (31, 31+20, s)-Nets in Base 7

(31, 31+20, 113)-Net over F₇ — Constructive and digital

Digital (31, 51, 113)-net over F₇, using

(u,Â u+v)-construction [i] based on
1. digital (1, 11, 13)-net over F₇, using
  - 2 times m-reduction [i] based on digital (1, 13, 13)-net over F₇, using
    - a shift-net [i]
2. digital (20, 40, 100)-net over F₇, using
  - trace code for nets [i] based on digital (0, 20, 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]

(31, 31+20, 255)-Net over F₇ — Digital

Digital (31, 51, 255)-net over F₇, using

Gilbertâ€“VarÅ¡amov bound [i]

(31, 31+20, 15404)-Net in Base 7 — Upper bound on s

There is no (31, 51, 15405)-net in base 7, because

the generalized Rao bound for nets shows that 7^mÂ â‰¥ 12 592281 036400 354271 182212 736608 623244 623209 > 7⁵¹ [i]