MinT - Best Known (37−22, 37, s)-Nets in Base 16

Best Known (37−22, 37, s)-Nets in Base 16

(37−22, 37, 66)-Net over F₁₆ — Constructive and digital

Digital (15, 37, 66)-net over F₁₆, using

(u,Â u+v)-construction [i] based on
1. digital (2, 13, 33)-net over F₁₆, using
  - net from sequence [i] based on digital (2, 32)-sequence over F₁₆, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₆ with g(F) = 2 and N(F) ≥ 33, using
      - a function field by SÃ©mirat [i]
2. digital (2, 24, 33)-net over F₁₆, using
  - net from sequence [i] based on digital (2, 32)-sequence over F₁₆ (see above)

(37−22, 37, 98)-Net in Base 16 — Constructive

(15, 37, 98)-net in base 16, using

3 times m-reduction [i] based on (15, 40, 98)-net in base 16, using
- base change [i] based on digital (7, 32, 98)-net over F₃₂, using
  - net from sequence [i] based on digital (7, 97)-sequence over F₃₂, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃₂ with g(F) = 7 and N(F) ≥ 98, using
      - a function field by SÃ©mirat [i]

(37−22, 37, 98)-Net over F₁₆ — Digital

Digital (15, 37, 98)-net over F₁₆, using

net from sequence [i] based on digital (15, 97)-sequence over F₁₆, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₆ with g(F) = 15 and N(F) ≥ 98, using
  - a curve from the manYPoints database [i]

(37−22, 37, 3668)-Net in Base 16 — Upper bound on s

There is no (15, 37, 3669)-net in base 16, because

the generalized Rao bound for nets shows that 16^mÂ â‰¥ 357 543673 288472 479741 004420 365036 511741 544136 > 16³⁷ [i]