MinT - Best Known (32, 129, s)-Nets in Base 16

Best Known (32, 129, s)-Nets in Base 16

(32, 129, 65)-Net over F₁₆ — Constructive and digital

Digital (32, 129, 65)-net over F₁₆, using

t-expansion [i] based on digital (6, 129, 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]

(32, 129, 76)-Net in Base 16 — Constructive

(32, 129, 76)-net in base 16, using

t-expansion [i] based on (31, 129, 76)-net in base 16, using
- 1 times m-reduction [i] based on (31, 130, 76)-net in base 16, using
  - base change [i] based on digital (5, 104, 76)-net over F₃₂, using
    - net from sequence [i] based on digital (5, 75)-sequence over F₃₂, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃₂ with g(F) = 5 and N(F) ≥ 76, using
        a function field by SÃ©mirat [i]

(32, 129, 168)-Net over F₁₆ — Digital

Digital (32, 129, 168)-net over F₁₆, using

t-expansion [i] based on digital (31, 129, 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]

(32, 129, 2004)-Net in Base 16 — Upper bound on s

There is no (32, 129, 2005)-net in base 16, because

1 times m-reduction [i] would yield (32, 128, 2005)-net in base 16, but
- the generalized Rao bound for nets shows that 16^mÂ â‰¥ 13640 967878 024699 171833 257955 273123 908840 541982 831376 768830 832905 111508 776528 966005 466975 243498 001529 233088 624872 803147 940890 380849 873124 967801 340519 318726 > 16¹²⁸ [i]