MinT - Best Known (81−49, 81, s)-Nets in Base 16

Best Known (81−49, 81, s)-Nets in Base 16

(81−49, 81, 98)-Net over F₁₆ — Constructive and digital

Digital (32, 81, 98)-net over F₁₆, using

(u,Â u+v)-construction [i] based on
1. digital (2, 26, 33)-net over F₁₆, using
  - net from sequence [i] based on digital (2, 32)-sequence over F₁₆, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₆ with g(F) = 2 and N(F) ≥ 33, 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]

(81−49, 81, 128)-Net in Base 16 — Constructive

(32, 81, 128)-net in base 16, using

base change [i] based on digital (5, 54, 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]

(81−49, 81, 168)-Net over F₁₆ — Digital

Digital (32, 81, 168)-net over F₁₆, using

t-expansion [i] based on digital (31, 81, 168)-net over F₁₆, using
- net from sequence [i] based on digital (31, 167)-sequence over F₁₆, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₆ with g(F) = 31 and N(F) ≥ 168, using
    - a curve from the manYPoints database [i]

(81−49, 81, 6732)-Net in Base 16 — Upper bound on s

There is no (32, 81, 6733)-net in base 16, because

1 times m-reduction [i] would yield (32, 80, 6733)-net in base 16, but
- the generalized Rao bound for nets shows that 16^mÂ â‰¥ 2 141613 822231 460348 930291 497418 020150 128903 159440 105872 369189 892561 283524 565115 661151 784053 261356 > 16⁸⁰ [i]