MinT - Best Known (81−43, 81, s)-Nets in Base 16

Best Known (81−43, 81, s)-Nets in Base 16

(81−43, 81, 130)-Net over F₁₆ — Constructive and digital

Digital (38, 81, 130)-net over F₁₆, using

9 times m-reduction [i] based on digital (38, 90, 130)-net over F₁₆, using
- (u,Â u+v)-construction [i] based on
  1. digital (6, 32, 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, 58, 65)-net over F₁₆, using
    - net from sequence [i] based on digital (6, 64)-sequence over F₁₆ (see above)

(81−43, 81, 192)-Net in Base 16 — Constructive

(38, 81, 192)-net in base 16, using

base change [i] based on (11, 54, 192)-net in base 64, using
- 2 times m-reduction [i] based on (11, 56, 192)-net in base 64, using
  - base change [i] based on digital (3, 48, 192)-net over F₁₂₈, using
    - net from sequence [i] based on digital (3, 191)-sequence over F₁₂₈, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₂₈ with g(F) = 3 and N(F) ≥ 192, using
        a function field by SÃ©mirat [i]

(81−43, 81, 224)-Net over F₁₆ — Digital

Digital (38, 81, 224)-net over F₁₆, using

Gilbertâ€“VarÅ¡amov bound [i]

(81−43, 81, 225)-Net in Base 16

(38, 81, 225)-net in base 16, using

3 times m-reduction [i] based on (38, 84, 225)-net in base 16, using
- base change [i] based on digital (10, 56, 225)-net over F₆₄, using
  - net from sequence [i] based on digital (10, 224)-sequence over F₆₄, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 10 and N(F) ≥ 225, using
      - a curve from the manYPoints database [i]

(81−43, 81, 22351)-Net in Base 16 — Upper bound on s

There is no (38, 81, 22352)-net in base 16, because

1 times m-reduction [i] would yield (38, 80, 22352)-net in base 16, but
- the generalized Rao bound for nets shows that 16^mÂ â‰¥ 2 137304 450779 462175 020354 326451 357012 645396 777474 584879 831053 947137 835297 228614 312246 509889 190756 > 16⁸⁰ [i]