MinT - Best Known (56−27, 56, s)-Nets in Base 7

Best Known (56−27, 56, s)-Nets in Base 7

(56−27, 56, 102)-Net over F₇ — Constructive and digital

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]

(56−27, 56, 128)-Net over F₇ — Digital

Digital (29, 56, 128)-net over F₇, using

trace code for nets [i] based on digital (1, 28, 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]

(56−27, 56, 3545)-Net in Base 7 — Upper bound on s

There is no (29, 56, 3546)-net in base 7, because

1 times m-reduction [i] would yield (29, 55, 3546)-net in base 7, but
- the generalized Rao bound for nets shows that 7^mÂ â‰¥ 30287 770341 299269 410873 638575 252674 741279 426373 > 7⁵⁵ [i]