MinT - Best Known (42−19, 42, s)-Nets in Base 7

Best Known (42−19, 42, s)-Nets in Base 7

(42−19, 42, 104)-Net over F₇ — Constructive and digital

Digital (23, 42, 104)-net over F₇, using

trace code for nets [i] based on digital (2, 21, 52)-net over F₄₉, using
- net from sequence [i] based on digital (2, 51)-sequence over F₄₉, using
  - Niederreiter sequence [i]

(42−19, 42, 156)-Net over F₇ — Digital

Digital (23, 42, 156)-net over F₇, using

trace code for nets [i] based on digital (2, 21, 78)-net over F₄₉, using
- net from sequence [i] based on digital (2, 77)-sequence over F₄₉, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄₉ with g(F) = 2 and N(F) ≥ 78, using
    - sharp bound on the number of rational points on curves with genus 2 [i]

(42−19, 42, 4886)-Net in Base 7 — Upper bound on s

There is no (23, 42, 4887)-net in base 7, because

1 times m-reduction [i] would yield (23, 41, 4887)-net in base 7, but
- the generalized Rao bound for nets shows that 7^mÂ â‰¥ 44604 024215 317507 544665 929778 020187 > 7⁴¹ [i]