MinT - Best Known (65, 65+27, s)-Nets in Base 7

Best Known (65, 65+27, s)-Nets in Base 7

(65, 65+27, 215)-Net over F₇ — Constructive and digital

Digital (65, 92, 215)-net over F₇, using

generalized (u,Â u+v)-construction [i] based on
1. digital (1, 10, 13)-net over F₇, using
  - 3 times m-reduction [i] based on digital (1, 13, 13)-net over F₇, using
    - a shift-net [i]
2. digital (13, 26, 100)-net over F₇, using
  - trace code for nets [i] based on digital (0, 13, 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 (29, 56, 102)-net over F₇, using
  - trace code for nets [i] based on digital (1, 28, 51)-net over F₄₉, using
    - net from sequence [i] based on digital (1, 50)-sequence over F₄₉, using
      - Niederreiter sequence [i]

(65, 65+27, 1733)-Net over F₇ — Digital

Digital (65, 92, 1733)-net over F₇, using

Gilbertâ€“VarÅ¡amov bound [i]

(65, 65+27, 777923)-Net in Base 7 — Upper bound on s

There is no (65, 92, 777924)-net in base 7, because

1 times m-reduction [i] would yield (65, 91, 777924)-net in base 7, but
- the generalized Rao bound for nets shows that 7^mÂ â‰¥ 80154 594534 802636 274799 907200 996638 286811 103739 354062 547908 373918 097012 902073 > 7⁹¹ [i]