MinT - Best Known (83−59, 83, s)-Nets in Base 16

Best Known (83−59, 83, s)-Nets in Base 16

(83−59, 83, 65)-Net over F₁₆ — Constructive and digital

Digital (24, 83, 65)-net over F₁₆, using

t-expansion [i] based on digital (6, 83, 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]

(83−59, 83, 98)-Net in Base 16 — Constructive

(24, 83, 98)-net in base 16, using

2 times m-reduction [i] based on (24, 85, 98)-net in base 16, using
- base change [i] based on digital (7, 68, 98)-net over F₃₂, using
  - net from sequence [i] based on digital (7, 97)-sequence over F₃₂, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃₂ with g(F) = 7 and N(F) ≥ 98, using
      - a function field by SÃ©mirat [i]

(83−59, 83, 129)-Net over F₁₆ — Digital

Digital (24, 83, 129)-net over F₁₆, using

t-expansion [i] based on digital (19, 83, 129)-net over F₁₆, using
- net from sequence [i] based on digital (19, 128)-sequence over F₁₆, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₆ with g(F) = 19 and N(F) ≥ 129, using
    - a curve from the manYPoints database [i]

(83−59, 83, 1960)-Net in Base 16 — Upper bound on s

There is no (24, 83, 1961)-net in base 16, because

1 times m-reduction [i] would yield (24, 82, 1961)-net in base 16, but
- the generalized Rao bound for nets shows that 16^mÂ â‰¥ 554 205763 246794 018694 507405 718136 129480 860056 613113 729738 747146 747494 374462 865719 029224 647144 383136 > 16⁸² [i]