MinT - Best Known (28, 81, s)-Nets in Base 16

Best Known (28, 81, s)-Nets in Base 16

(28, 81, 65)-Net over F₁₆ — Constructive and digital

Digital (28, 81, 65)-net over F₁₆, using

t-expansion [i] based on digital (6, 81, 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]

(28, 81, 120)-Net in Base 16 — Constructive

(28, 81, 120)-net in base 16, using

4 times m-reduction [i] based on (28, 85, 120)-net in base 16, using
- base change [i] based on digital (11, 68, 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]

(28, 81, 156)-Net over F₁₆ — Digital

Digital (28, 81, 156)-net over F₁₆, using

t-expansion [i] based on digital (27, 81, 156)-net over F₁₆, using
- net from sequence [i] based on digital (27, 155)-sequence over F₁₆, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₆ with g(F) = 27 and N(F) ≥ 156, using
    - a curve from the manYPoints database [i]

(28, 81, 3551)-Net in Base 16 — Upper bound on s

There is no (28, 81, 3552)-net in base 16, because

1 times m-reduction [i] would yield (28, 80, 3552)-net in base 16, but
- the generalized Rao bound for nets shows that 16^mÂ â‰¥ 2 138759 249657 148767 659158 948086 626347 886321 352271 863794 068029 895930 226853 909426 365693 952991 751531 > 16⁸⁰ [i]