MinT - Best Known (17, 42, s)-Nets in Base 16

Best Known (17, 42, s)-Nets in Base 16

(17, 42, 71)-Net over F₁₆ — Constructive and digital

Digital (17, 42, 71)-net over F₁₆, using

(u,Â u+v)-construction [i] based on
1. digital (2, 14, 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, 28, 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]

(17, 42, 104)-Net in Base 16 — Constructive

(17, 42, 104)-net in base 16, using

base change [i] based on digital (3, 28, 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]

(17, 42, 112)-Net over F₁₆ — Digital

Digital (17, 42, 112)-net over F₁₆, using

net from sequence [i] based on digital (17, 111)-sequence over F₁₆, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₆ with g(F) = 17 and N(F) ≥ 112, using
  - a curve from the manYPoints database [i]

(17, 42, 113)-Net in Base 16

(17, 42, 113)-net in base 16, using

base change [i] based on digital (3, 28, 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]

(17, 42, 4578)-Net in Base 16 — Upper bound on s

There is no (17, 42, 4579)-net in base 16, because

1 times m-reduction [i] would yield (17, 41, 4579)-net in base 16, but
- the generalized Rao bound for nets shows that 16^mÂ â‰¥ 23 397452 203521 199522 426452 576840 903279 106596 097096 > 16⁴¹ [i]