MinT - Best Known (111−85, 111, s)-Nets in Base 16

Best Known (111−85, 111, s)-Nets in Base 16

(111−85, 111, 65)-Net over F₁₆ — Constructive and digital

Digital (26, 111, 65)-net over F₁₆, using

t-expansion [i] based on digital (6, 111, 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]

(111−85, 111, 66)-Net in Base 16 — Constructive

(26, 111, 66)-net in base 16, using

t-expansion [i] based on (25, 111, 66)-net in base 16, using
- net from sequence [i] based on (25, 65)-sequence in base 16, using
  - base expansion [i] based on digital (50, 65)-sequence over F₄, using
    - t-expansion [i] based on digital (49, 65)-sequence over F₄, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 49 and N(F) ≥ 66, using
        T₆ from the second tower of function fields by GarcÃa and Stichtenoth over F₄ [i]

(111−85, 111, 150)-Net over F₁₆ — Digital

Digital (26, 111, 150)-net over F₁₆, using

net from sequence [i] based on digital (26, 149)-sequence over F₁₆, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₆ with g(F) = 26 and N(F) ≥ 150, using
  - a curve from the manYPoints database [i]

(111−85, 111, 1544)-Net in Base 16 — Upper bound on s

There is no (26, 111, 1545)-net in base 16, because

1 times m-reduction [i] would yield (26, 110, 1545)-net in base 16, but
- the generalized Rao bound for nets shows that 16^mÂ â‰¥ 2 852226 750130 472821 929713 189532 333586 965584 881770 724392 588274 914500 885553 097402 630191 533830 335828 513048 372757 092571 948347 358379 934976 > 16¹¹⁰ [i]