MinT - Best Known (60−37, 60, s)-Nets in Base 16

Best Known (60−37, 60, s)-Nets in Base 16

(60−37, 60, 71)-Net over F₁₆ — Constructive and digital

Digital (23, 60, 71)-net over F₁₆, using

(u,Â u+v)-construction [i] based on
1. digital (2, 20, 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 (3, 40, 38)-net over F₁₆, using
  - net from sequence [i] based on digital (3, 37)-sequence over F₁₆, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₆ with g(F) = 3 and N(F) ≥ 38, using
      - a function field by SÃ©mirat [i]

(60−37, 60, 120)-Net in Base 16 — Constructive

(23, 60, 120)-net in base 16, using

base change [i] based on digital (11, 48, 120)-net over F₃₂, using
- net from sequence [i] based on digital (11, 119)-sequence over F₃₂, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃₂ with g(F) = 11 and N(F) ≥ 120, using
    - a function field by SÃ©mirat [i]

(60−37, 60, 129)-Net over F₁₆ — Digital

Digital (23, 60, 129)-net over F₁₆, using

t-expansion [i] based on digital (19, 60, 129)-net over F₁₆, using
- net from sequence [i] based on digital (19, 128)-sequence over F₁₆, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₆ with g(F) = 19 and N(F) ≥ 129, using
    - a curve from the manYPoints database [i]

(60−37, 60, 4445)-Net in Base 16 — Upper bound on s

There is no (23, 60, 4446)-net in base 16, because

1 times m-reduction [i] would yield (23, 59, 4446)-net in base 16, but
- the generalized Rao bound for nets shows that 16^mÂ â‰¥ 110606 086705 309797 127324 735836 839559 389651 749668 807940 602099 066133 965346 > 16⁵⁹ [i]