MinT - Best Known (15−10, 15, s)-Nets in Base 81

Best Known (15−10, 15, s)-Nets in Base 81

(15−10, 15, 164)-Net over F₈₁ — Constructive and digital

Digital (5, 15, 164)-net over F₈₁, using

(u,Â u+v)-construction [i] based on
1. digital (0, 5, 82)-net over F₈₁, using
  - net from sequence [i] based on digital (0, 81)-sequence over F₈₁, using
    - generalized Faure sequence [i]
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈₁ with g(F) = 0 and N(F) ≥ 82, using
      - the rational function field F₈₁(x) [i]
    - Niederreiter sequence [i]
2. digital (0, 10, 82)-net over F₈₁, using
  - net from sequence [i] based on digital (0, 81)-sequence over F₈₁ (see above)

(15−10, 15, 167)-Net over F₈₁ — Digital

Digital (5, 15, 167)-net over F₈₁, using

net from sequence [i] based on digital (5, 166)-sequence over F₈₁, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈₁ with g(F) = 5 and N(F) ≥ 167, using
  - a curve from the manYPoints database [i]

(15−10, 15, 17304)-Net in Base 81 — Upper bound on s

There is no (5, 15, 17305)-net in base 81, because

the generalized Rao bound for nets shows that 81^mÂ â‰¥ 42401 916929 324780 720529 482001 > 81¹⁵ [i]