MinT - Best Known (88−29, 88, s)-Nets in Base 7

Best Known (88−29, 88, s)-Nets in Base 7

(88−29, 88, 202)-Net over F₇ — Constructive and digital

Digital (59, 88, 202)-net over F₇, using

(u,Â u+v)-construction [i] based on
1. digital (14, 28, 100)-net over F₇, using
  - trace code for nets [i] based on digital (0, 14, 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 (31, 60, 102)-net over F₇, using
  - trace code for nets [i] based on digital (1, 30, 51)-net over F₄₉, using
    - net from sequence [i] based on digital (1, 50)-sequence over F₄₉, using
      - Niederreiter sequence [i]

(88−29, 88, 867)-Net over F₇ — Digital

Digital (59, 88, 867)-net over F₇, using

Gilbertâ€“VarÅ¡amov bound [i]

(88−29, 88, 179872)-Net in Base 7 — Upper bound on s

There is no (59, 88, 179873)-net in base 7, because

1 times m-reduction [i] would yield (59, 87, 179873)-net in base 7, but
- the generalized Rao bound for nets shows that 7^mÂ â‰¥ 33 383465 720001 861814 669280 095826 638899 724023 701119 062225 217775 461523 998609 > 7⁸⁷ [i]