MinT - Best Known (32, 32+57, s)-Nets in Base 16

Best Known (32, 32+57, s)-Nets in Base 16

(32, 32+57, 66)-Net over F₁₆ — Constructive and digital

Digital (32, 89, 66)-net over F₁₆, using

(u,Â u+v)-construction [i] based on
1. digital (2, 30, 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, 59, 33)-net over F₁₆, using
  - net from sequence [i] based on digital (2, 32)-sequence over F₁₆ (see above)

(32, 32+57, 120)-Net in Base 16 — Constructive

(32, 89, 120)-net in base 16, using

16 times m-reduction [i] based on (32, 105, 120)-net in base 16, using
- base change [i] based on digital (11, 84, 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]

(32, 32+57, 168)-Net over F₁₆ — Digital

Digital (32, 89, 168)-net over F₁₆, using

t-expansion [i] based on digital (31, 89, 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, 32+57, 4569)-Net in Base 16 — Upper bound on s

There is no (32, 89, 4570)-net in base 16, because

1 times m-reduction [i] would yield (32, 88, 4570)-net in base 16, but
- the generalized Rao bound for nets shows that 16^mÂ â‰¥ 9222 613244 206457 593351 349277 883284 798752 757639 430893 918332 668164 678324 625192 044751 286704 722371 987000 681776 > 16⁸⁸ [i]