MinT - Best Known (66−29, 66, s)-Nets in Base 7

Best Known (66−29, 66, s)-Nets in Base 7

(66−29, 66, 108)-Net over F₇ — Constructive and digital

Digital (37, 66, 108)-net over F₇, using

trace code for nets [i] based on digital (4, 33, 54)-net over F₄₉, using
- net from sequence [i] based on digital (4, 53)-sequence over F₄₉, using
  - Niederreiter sequence [i]

(66−29, 66, 185)-Net over F₇ — Digital

Digital (37, 66, 185)-net over F₇, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(7⁶⁶, 185, F₇, 2, 29) (dual of [(185, 2), 304, 30]-NRT-code), using
- 1 times NRT-code embedding in larger space [i] based on linear OOA(7⁶⁴, 184, F₇, 2, 29) (dual of [(184, 2), 304, 30]-NRT-code), using
  - extracting embedded OOA [i] based on digital (35, 64, 184)-net over F₇, using
    - trace code for nets [i] based on digital (3, 32, 92)-net over F₄₉, using
      - net from sequence [i] based on digital (3, 91)-sequence over F₄₉, using
        Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄₉ with g(F) = 3 and N(F) ≥ 92, using
        a curve from the manYPoints database [i]

(66−29, 66, 8443)-Net in Base 7 — Upper bound on s

There is no (37, 66, 8444)-net in base 7, because

1 times m-reduction [i] would yield (37, 65, 8444)-net in base 7, but
- the generalized Rao bound for nets shows that 7^mÂ â‰¥ 8 545534 672999 038184 909205 752065 647853 347886 125747 112073 > 7⁶⁵ [i]