MinT - Best Known (81−37, 81, s)-Nets in Base 7

Best Known (81−37, 81, s)-Nets in Base 7

(81−37, 81, 106)-Net over F₇ — Constructive and digital

Digital (44, 81, 106)-net over F₇, using

1 times m-reduction [i] based on digital (44, 82, 106)-net over F₇, using
- trace code for nets [i] based on digital (3, 41, 53)-net over F₄₉, using
  - net from sequence [i] based on digital (3, 52)-sequence over F₄₉, using
    - Niederreiter sequence [i]

(81−37, 81, 184)-Net over F₇ — Digital

Digital (44, 81, 184)-net over F₇, using

1 times m-reduction [i] based on digital (44, 82, 184)-net over F₇, using
- trace code for nets [i] based on digital (3, 41, 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]

(81−37, 81, 7165)-Net in Base 7 — Upper bound on s

There is no (44, 81, 7166)-net in base 7, because

1 times m-reduction [i] would yield (44, 80, 7166)-net in base 7, but
- the generalized Rao bound for nets shows that 7^mÂ â‰¥ 40 540822 504844 123071 152952 497585 472520 776061 724606 432938 740461 894829 > 7⁸⁰ [i]