MinT - Best Known (25, 25+77, s)-Nets in Base 16

Best Known (25, 25+77, s)-Nets in Base 16

(25, 25+77, 65)-Net over F₁₆ — Constructive and digital

Digital (25, 102, 65)-net over F₁₆, using

t-expansion [i] based on digital (6, 102, 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]

(25, 25+77, 66)-Net in Base 16 — Constructive

(25, 102, 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]

(25, 25+77, 144)-Net over F₁₆ — Digital

Digital (25, 102, 144)-net over F₁₆, using

net from sequence [i] based on digital (25, 143)-sequence over F₁₆, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₆ with g(F) = 25 and N(F) ≥ 144, using
  - a curve from the manYPoints database [i]

(25, 25+77, 1568)-Net in Base 16 — Upper bound on s

There is no (25, 102, 1569)-net in base 16, because

1 times m-reduction [i] would yield (25, 101, 1569)-net in base 16, but
- the generalized Rao bound for nets shows that 16^mÂ â‰¥ 42 234561 460349 863638 342166 379547 620072 695319 101643 327584 349365 367382 614255 969428 123843 441561 562437 799830 709585 038360 819456 > 16¹⁰¹ [i]