MinT - Best Known (21, 21+19, s)-Nets in Base 7

Best Known (21, 21+19, s)-Nets in Base 7

(21, 21+19, 102)-Net over F₇ — Constructive and digital

Digital (21, 40, 102)-net over F₇, using

trace code for nets [i] based on digital (1, 20, 51)-net over F₄₉, using
- net from sequence [i] based on digital (1, 50)-sequence over F₄₉, using
  - Niederreiter sequence [i]

(21, 21+19, 128)-Net over F₇ — Digital

Digital (21, 40, 128)-net over F₇, using

trace code for nets [i] based on digital (1, 20, 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]

(21, 21+19, 3169)-Net in Base 7 — Upper bound on s

There is no (21, 40, 3170)-net in base 7, because

1 times m-reduction [i] would yield (21, 39, 3170)-net in base 7, but
- the generalized Rao bound for nets shows that 7^mÂ â‰¥ 911 807733 868272 844936 659792 827965 > 7³⁹ [i]