MinT - Best Known (32, 73, s)-Nets in Base 16

Best Known (32, 73, s)-Nets in Base 16

(32, 73, 130)-Net over F₁₆ — Constructive and digital

Digital (32, 73, 130)-net over F₁₆, using

(u,Â u+v)-construction [i] based on
1. digital (6, 26, 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]
2. digital (6, 47, 65)-net over F₁₆, using
  - net from sequence [i] based on digital (6, 64)-sequence over F₁₆ (see above)

(32, 73, 168)-Net over F₁₆ — Digital

Digital (32, 73, 168)-net over F₁₆, using

t-expansion [i] based on digital (31, 73, 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, 73, 177)-Net in Base 16 — Constructive

(32, 73, 177)-net in base 16, using

2 times m-reduction [i] based on (32, 75, 177)-net in base 16, using
- base change [i] based on digital (7, 50, 177)-net over F₆₄, using
  - net from sequence [i] based on digital (7, 176)-sequence over F₆₄, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 7 and N(F) ≥ 177, using
      - a function field by GarcÃa and Quoos [i]

(32, 73, 11957)-Net in Base 16 — Upper bound on s

There is no (32, 73, 11958)-net in base 16, because

1 times m-reduction [i] would yield (32, 72, 11958)-net in base 16, but
- the generalized Rao bound for nets shows that 16^mÂ â‰¥ 497 433610 648927 625263 817594 416576 352378 572644 643891 425965 251366 526960 871994 321101 982651 > 16⁷² [i]