MinT - Best Known (32, 32+51, s)-Nets in Base 16

Best Known (32, 32+51, s)-Nets in Base 16

(32, 32+51, 89)-Net over F₁₆ — Constructive and digital

Digital (32, 83, 89)-net over F₁₆, using

(u,Â u+v)-construction [i] based on
1. digital (1, 26, 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, 57, 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, 32+51, 120)-Net in Base 16 — Constructive

(32, 83, 120)-net in base 16, using

22 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+51, 168)-Net over F₁₆ — Digital

Digital (32, 83, 168)-net over F₁₆, using

t-expansion [i] based on digital (31, 83, 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+51, 6026)-Net in Base 16 — Upper bound on s

There is no (32, 83, 6027)-net in base 16, because

1 times m-reduction [i] would yield (32, 82, 6027)-net in base 16, but
- the generalized Rao bound for nets shows that 16^mÂ â‰¥ 547 953249 199765 884788 489217 786100 146753 523282 210206 878394 800973 961851 031045 328571 540728 257932 231376 > 16⁸² [i]