MinT - Best Known (65−37, 65, s)-Nets in Base 16

Best Known (65−37, 65, s)-Nets in Base 16

(65−37, 65, 110)-Net over F₁₆ — Constructive and digital

Digital (28, 65, 110)-net over F₁₆, using

(u,Â u+v)-construction [i] based on
1. digital (4, 22, 45)-net over F₁₆, using
  - net from sequence [i] based on digital (4, 44)-sequence over F₁₆, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₆ with g(F) = 4 and N(F) ≥ 45, using
      - a function field by GarcÃa and Quoos [i]
2. digital (6, 43, 65)-net over F₁₆, using
  - net from sequence [i] based on digital (6, 64)-sequence over F₁₆, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₆ with g(F) = 6 and N(F) ≥ 65, using
      - the Hermitian function field over F₁₆ [i]

(65−37, 65, 128)-Net in Base 16 — Constructive

(28, 65, 128)-net in base 16, using

4 times m-reduction [i] based on (28, 69, 128)-net in base 16, using
- base change [i] based on digital (5, 46, 128)-net over F₆₄, using
  - net from sequence [i] based on digital (5, 127)-sequence over F₆₄, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 5 and N(F) ≥ 128, using
      - a function field by SÃ©mirat [i]

(65−37, 65, 156)-Net over F₁₆ — Digital

Digital (28, 65, 156)-net over F₁₆, using

t-expansion [i] based on digital (27, 65, 156)-net over F₁₆, using
- net from sequence [i] based on digital (27, 155)-sequence over F₁₆, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₆ with g(F) = 27 and N(F) ≥ 156, using
    - a curve from the manYPoints database [i]

(65−37, 65, 161)-Net in Base 16

(28, 65, 161)-net in base 16, using

1 times m-reduction [i] based on (28, 66, 161)-net in base 16, using
- base change [i] based on digital (6, 44, 161)-net over F₆₄, using
  - net from sequence [i] based on digital (6, 160)-sequence over F₆₄, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 6 and N(F) ≥ 161, using
      - a curve from the manYPoints database [i]

(65−37, 65, 9614)-Net in Base 16 — Upper bound on s

There is no (28, 65, 9615)-net in base 16, because

1 times m-reduction [i] would yield (28, 64, 9615)-net in base 16, but
- the generalized Rao bound for nets shows that 16^mÂ â‰¥ 115949 589858 421326 832545 228121 442891 882691 513532 028614 476603 091114 507505 615176 > 16⁶⁴ [i]