MinT - Best Known (72−41, 72, s)-Nets in Base 16

Best Known (72−41, 72, s)-Nets in Base 16

(72−41, 72, 114)-Net over F₁₆ — Constructive and digital

Digital (31, 72, 114)-net over F₁₆, using

(u,Â u+v)-construction [i] based on
1. digital (5, 25, 49)-net over F₁₆, using
  - net from sequence [i] based on digital (5, 48)-sequence over F₁₆, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₆ with g(F) = 5 and N(F) ≥ 49, using
      - a function field by SÃ©mirat [i]
2. digital (6, 47, 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]

(72−41, 72, 168)-Net over F₁₆ — Digital

Digital (31, 72, 168)-net over F₁₆, using

net from sequence [i] based on digital (31, 167)-sequence over F₁₆, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₆ with g(F) = 31 and N(F) ≥ 168, using
  - a curve from the manYPoints database [i]

(72−41, 72, 177)-Net in Base 16 — Constructive

(31, 72, 177)-net in base 16, using

base change [i] based on digital (7, 48, 177)-net over F₆₄, using
- net from sequence [i] based on digital (7, 176)-sequence over F₆₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 7 and N(F) ≥ 177, using
    - a function field by GarcÃa and Quoos [i]

(72−41, 72, 10408)-Net in Base 16 — Upper bound on s

There is no (31, 72, 10409)-net in base 16, because

1 times m-reduction [i] would yield (31, 71, 10409)-net in base 16, but
- the generalized Rao bound for nets shows that 16^mÂ â‰¥ 31 110973 607211 821313 349408 087304 702946 757554 276452 977025 128090 737721 177058 780589 987076 > 16⁷¹ [i]