MinT - Best Known (16, 39, s)-Nets in Base 16

Best Known (16, 39, s)-Nets in Base 16

(16, 39, 71)-Net over F₁₆ — Constructive and digital

Digital (16, 39, 71)-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 (3, 26, 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]

(16, 39, 98)-Net over F₁₆ — Digital

Digital (16, 39, 98)-net over F₁₆, using

t-expansion [i] based on digital (15, 39, 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]

(16, 39, 104)-Net in Base 16 — Constructive

(16, 39, 104)-net in base 16, using

base change [i] based on digital (3, 26, 104)-net over F₆₄, using
- net from sequence [i] based on digital (3, 103)-sequence over F₆₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 3 and N(F) ≥ 104, using
    - a function field by SÃ©mirat [i]

(16, 39, 113)-Net in Base 16

(16, 39, 113)-net in base 16, using

base change [i] based on digital (3, 26, 113)-net over F₆₄, using
- net from sequence [i] based on digital (3, 112)-sequence over F₆₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 3 and N(F) ≥ 113, using
    - a curve from the manYPoints database [i]

(16, 39, 4721)-Net in Base 16 — Upper bound on s

There is no (16, 39, 4722)-net in base 16, because

1 times m-reduction [i] would yield (16, 38, 4722)-net in base 16, but
- the generalized Rao bound for nets shows that 16^mÂ â‰¥ 5715 166967 499757 852584 786282 466582 260443 931256 > 16³⁸ [i]