MinT - Best Known (68−32, 68, s)-Nets in Base 7

Best Known (68−32, 68, s)-Nets in Base 7

(68−32, 68, 104)-Net over F₇ — Constructive and digital

Digital (36, 68, 104)-net over F₇, using

trace code for nets [i] based on digital (2, 34, 52)-net over F₄₉, using
- net from sequence [i] based on digital (2, 51)-sequence over F₄₉, using
  - Niederreiter sequence [i]

(68−32, 68, 156)-Net over F₇ — Digital

Digital (36, 68, 156)-net over F₇, using

trace code for nets [i] based on digital (2, 34, 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]

(68−32, 68, 4416)-Net in Base 7 — Upper bound on s

There is no (36, 68, 4417)-net in base 7, because

the generalized Rao bound for nets shows that 7^mÂ â‰¥ 2936 312912 339636 120295 491782 620722 965011 252110 154224 999809 > 7⁶⁸ [i]