MinT - Best Known (26, 26+87, s)-Nets in Base 16

Best Known (26, 26+87, s)-Nets in Base 16

(26, 26+87, 65)-Net over F₁₆ — Constructive and digital

Digital (26, 113, 65)-net over F₁₆, using

t-expansion [i] based on digital (6, 113, 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]

(26, 26+87, 66)-Net in Base 16 — Constructive

(26, 113, 66)-net in base 16, using

t-expansion [i] based on (25, 113, 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]

(26, 26+87, 150)-Net over F₁₆ — Digital

Digital (26, 113, 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]

(26, 26+87, 1516)-Net in Base 16 — Upper bound on s

There is no (26, 113, 1517)-net in base 16, because

1 times m-reduction [i] would yield (26, 112, 1517)-net in base 16, but
- the generalized Rao bound for nets shows that 16^mÂ â‰¥ 730 414296 704048 864002 061447 027883 369883 268070 121293 122262 877746 442545 080542 541591 203012 952707 053510 420206 961166 194533 728239 281319 437116 > 16¹¹² [i]