MinT - Best Known (90−54, 90, s)-Nets in Base 16

Best Known (90−54, 90, s)-Nets in Base 16

(90−54, 90, 103)-Net over F₁₆ — Constructive and digital

Digital (36, 90, 103)-net over F₁₆, using

(u,Â u+v)-construction [i] based on
1. digital (3, 30, 38)-net over F₁₆, using
  - net from sequence [i] based on digital (3, 37)-sequence over F₁₆, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₆ with g(F) = 3 and N(F) ≥ 38, using
      - a function field by SÃ©mirat [i]
2. digital (6, 60, 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]

(90−54, 90, 128)-Net in Base 16 — Constructive

(36, 90, 128)-net in base 16, using

3 times m-reduction [i] based on (36, 93, 128)-net in base 16, using
- base change [i] based on digital (5, 62, 128)-net over F₆₄, using
  - net from sequence [i] based on digital (5, 127)-sequence over F₆₄, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 5 and N(F) ≥ 128, using
      - a function field by SÃ©mirat [i]

(90−54, 90, 193)-Net over F₁₆ — Digital

Digital (36, 90, 193)-net over F₁₆, using

t-expansion [i] based on digital (33, 90, 193)-net over F₁₆, using
- net from sequence [i] based on digital (33, 192)-sequence over F₁₆, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₆ with g(F) = 33 and N(F) ≥ 193, using
    - a curve from the manYPoints database [i]

(90−54, 90, 7502)-Net in Base 16 — Upper bound on s

There is no (36, 90, 7503)-net in base 16, because

the generalized Rao bound for nets shows that 16^mÂ â‰¥ 2 354618 043470 573862 786380 755446 544224 154986 344924 389929 893901 988481 530627 298372 786671 766530 438171 923419 668816 > 16⁹⁰ [i]