MinT - Best Known (30, 30+99, s)-Nets in Base 16

Best Known (30, 30+99, s)-Nets in Base 16

(30, 30+99, 65)-Net over F₁₆ — Constructive and digital

Digital (30, 129, 65)-net over F₁₆, using

t-expansion [i] based on digital (6, 129, 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]

(30, 30+99, 66)-Net in Base 16 — Constructive

(30, 129, 66)-net in base 16, using

t-expansion [i] based on (25, 129, 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]

(30, 30+99, 162)-Net over F₁₆ — Digital

Digital (30, 129, 162)-net over F₁₆, using

net from sequence [i] based on digital (30, 161)-sequence over F₁₆, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₆ with g(F) = 30 and N(F) ≥ 162, using
  - a curve from the manYPoints database [i]

(30, 30+99, 1753)-Net in Base 16 — Upper bound on s

There is no (30, 129, 1754)-net in base 16, because

1 times m-reduction [i] would yield (30, 128, 1754)-net in base 16, but
- the generalized Rao bound for nets shows that 16^mÂ â‰¥ 13454 227678 294308 874674 258812 617277 707712 383640 379643 835524 031776 961094 358208 156177 113361 527011 268336 713941 737960 268403 900415 561569 695849 271029 505732 089566 > 16¹²⁸ [i]