MinT - Best Known (74−44, 74, s)-Nets in Base 16

Best Known (74−44, 74, s)-Nets in Base 16

(74−44, 74, 98)-Net over F₁₆ — Constructive and digital

Digital (30, 74, 98)-net over F₁₆, using

(u,Â u+v)-construction [i] based on
1. digital (2, 24, 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 (6, 50, 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]

(74−44, 74, 128)-Net in Base 16 — Constructive

(30, 74, 128)-net in base 16, using

1 times m-reduction [i] based on (30, 75, 128)-net in base 16, using
- base change [i] based on digital (5, 50, 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]

(74−44, 74, 162)-Net over F₁₆ — Digital

Digital (30, 74, 162)-net over F₁₆, using

net from sequence [i] based on digital (30, 161)-sequence over F₁₆, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₆ with g(F) = 30 and N(F) ≥ 162, using
  - a curve from the manYPoints database [i]

(74−44, 74, 6764)-Net in Base 16 — Upper bound on s

There is no (30, 74, 6765)-net in base 16, because

the generalized Rao bound for nets shows that 16^mÂ â‰¥ 127647 171519 202480 823692 677116 310739 740806 171633 245410 982250 775091 346717 794679 693753 047576 > 16⁷⁴ [i]