MinT - Best Known (63, 63+31, s)-Nets in Base 7

Best Known (63, 63+31, s)-Nets in Base 7

(63, 63+31, 202)-Net over F₇ — Constructive and digital

Digital (63, 94, 202)-net over F₇, using

(u,Â u+v)-construction [i] based on
1. 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]
2. digital (33, 64, 102)-net over F₇, using
  - trace code for nets [i] based on digital (1, 32, 51)-net over F₄₉, using
    - net from sequence [i] based on digital (1, 50)-sequence over F₄₉, using
      - Niederreiter sequence [i]

(63, 63+31, 907)-Net over F₇ — Digital

Digital (63, 94, 907)-net over F₇, using

Gilbertâ€“VarÅ¡amov bound [i]

(63, 63+31, 185866)-Net in Base 7 — Upper bound on s

There is no (63, 94, 185867)-net in base 7, because

1 times m-reduction [i] would yield (63, 93, 185867)-net in base 7, but
- the generalized Rao bound for nets shows that 7^mÂ â‰¥ 3 927683 033704 349523 752671 239969 065979 003448 593354 099206 923518 259911 022307 545543 > 7⁹³ [i]