MinT - Best Known (45−21, 45, s)-Nets in Base 7

Best Known (45−21, 45, s)-Nets in Base 7

(45−21, 45, 102)-Net over F₇ — Constructive and digital

Digital (24, 45, 102)-net over F₇, using

1 times m-reduction [i] based on digital (24, 46, 102)-net over F₇, using
- trace code for nets [i] based on digital (1, 23, 51)-net over F₄₉, using
  - net from sequence [i] based on digital (1, 50)-sequence over F₄₉, using
    - Niederreiter sequence [i]

(45−21, 45, 128)-Net over F₇ — Digital

Digital (24, 45, 128)-net over F₇, using

1 times m-reduction [i] based on digital (24, 46, 128)-net over F₇, using
- trace code for nets [i] based on digital (1, 23, 64)-net over F₄₉, using
  - net from sequence [i] based on digital (1, 63)-sequence over F₄₉, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄₉ with g(F) = 1 and N(F) ≥ 64, using
      - sharp bound on the number of rational points on curves with genus 1 [i]

(45−21, 45, 3940)-Net in Base 7 — Upper bound on s

There is no (24, 45, 3941)-net in base 7, because

1 times m-reduction [i] would yield (24, 44, 3941)-net in base 7, but
- the generalized Rao bound for nets shows that 7^mÂ â‰¥ 15 296797 948369 672810 935466 070729 861209 > 7⁴⁴ [i]