MinT - Best Known (82−49, 82, s)-Nets in Base 16

Best Known (82−49, 82, s)-Nets in Base 16

(82−49, 82, 103)-Net over F₁₆ — Constructive and digital

Digital (33, 82, 103)-net over F₁₆, using

(u,Â u+v)-construction [i] based on
1. digital (3, 27, 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, 55, 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]

(82−49, 82, 128)-Net in Base 16 — Constructive

(33, 82, 128)-net in base 16, using

2 times m-reduction [i] based on (33, 84, 128)-net in base 16, using
- base change [i] based on digital (5, 56, 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]

(82−49, 82, 193)-Net over F₁₆ — Digital

Digital (33, 82, 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]

(82−49, 82, 7558)-Net in Base 16 — Upper bound on s

There is no (33, 82, 7559)-net in base 16, because

1 times m-reduction [i] would yield (33, 81, 7559)-net in base 16, but
- the generalized Rao bound for nets shows that 16^mÂ â‰¥ 34 250649 125610 345400 033719 063055 565542 184358 634403 638918 506783 815669 651107 422742 397216 485598 337616 > 16⁸¹ [i]