MinT - Best Known (74−45, 74, s)-Nets in Base 16

Best Known (74−45, 74, s)-Nets in Base 16

(74−45, 74, 89)-Net over F₁₆ — Constructive and digital

Digital (29, 74, 89)-net over F₁₆, using

(u,Â u+v)-construction [i] based on
1. digital (1, 23, 24)-net over F₁₆, using
  - net from sequence [i] based on digital (1, 23)-sequence over F₁₆, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₆ with g(F) = 1 and N(F) ≥ 24, using
      - a function field by SÃ©mirat [i]
2. digital (6, 51, 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]

(74−45, 74, 120)-Net in Base 16 — Constructive

(29, 74, 120)-net in base 16, using

16 times m-reduction [i] based on (29, 90, 120)-net in base 16, using
- base change [i] based on digital (11, 72, 120)-net over F₃₂, using
  - net from sequence [i] based on digital (11, 119)-sequence over F₃₂, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃₂ with g(F) = 11 and N(F) ≥ 120, using
      - a function field by SÃ©mirat [i]

(74−45, 74, 161)-Net over F₁₆ — Digital

Digital (29, 74, 161)-net over F₁₆, using

net from sequence [i] based on digital (29, 160)-sequence over F₁₆, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₆ with g(F) = 29 and N(F) ≥ 161, using
  - a curve from the manYPoints database [i]

(74−45, 74, 5961)-Net in Base 16 — Upper bound on s

There is no (29, 74, 5962)-net in base 16, because

1 times m-reduction [i] would yield (29, 73, 5962)-net in base 16, but
- the generalized Rao bound for nets shows that 16^mÂ â‰¥ 7961 828472 041764 810575 720796 152336 316526 209734 098373 713427 190999 778191 575323 185339 977136 > 16⁷³ [i]