MinT - Best Known (30, 83, s)-Nets in Base 16

Best Known (30, 83, s)-Nets in Base 16

(30, 83, 66)-Net over F₁₆ — Constructive and digital

Digital (30, 83, 66)-net over F₁₆, using

(u,Â u+v)-construction [i] based on
1. digital (2, 28, 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 (2, 55, 33)-net over F₁₆, using
  - net from sequence [i] based on digital (2, 32)-sequence over F₁₆ (see above)

(30, 83, 120)-Net in Base 16 — Constructive

(30, 83, 120)-net in base 16, using

12 times m-reduction [i] based on (30, 95, 120)-net in base 16, using
- base change [i] based on digital (11, 76, 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]

(30, 83, 162)-Net over F₁₆ — Digital

Digital (30, 83, 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]

(30, 83, 4399)-Net in Base 16 — Upper bound on s

There is no (30, 83, 4400)-net in base 16, because

1 times m-reduction [i] would yield (30, 82, 4400)-net in base 16, but
- the generalized Rao bound for nets shows that 16^mÂ â‰¥ 548 228336 694296 247144 994016 731003 251747 068561 009388 153831 010985 842356 248612 678382 205411 646975 794126 > 16⁸² [i]