MinT - Best Known (80−51, 80, s)-Nets in Base 16

Best Known (80−51, 80, s)-Nets in Base 16

(80−51, 80, 66)-Net over F₁₆ — Constructive and digital

Digital (29, 80, 66)-net over F₁₆, using

(u,Â u+v)-construction [i] based on
1. digital (2, 27, 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, 53, 33)-net over F₁₆, using
  - net from sequence [i] based on digital (2, 32)-sequence over F₁₆ (see above)

(80−51, 80, 120)-Net in Base 16 — Constructive

(29, 80, 120)-net in base 16, using

10 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]

(80−51, 80, 161)-Net over F₁₆ — Digital

Digital (29, 80, 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]

(80−51, 80, 4317)-Net in Base 16 — Upper bound on s

There is no (29, 80, 4318)-net in base 16, because

1 times m-reduction [i] would yield (29, 79, 4318)-net in base 16, but
- the generalized Rao bound for nets shows that 16^mÂ â‰¥ 134271 598977 032242 374166 065012 279886 425774 944002 111417 850161 228101 988798 578095 762845 281459 880501 > 16⁷⁹ [i]