MinT - Best Known (126−99, 126, s)-Nets in Base 16

Best Known (126−99, 126, s)-Nets in Base 16

(126−99, 126, 65)-Net over F₁₆ — Constructive and digital

Digital (27, 126, 65)-net over F₁₆, using

t-expansion [i] based on digital (6, 126, 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]

(126−99, 126, 66)-Net in Base 16 — Constructive

(27, 126, 66)-net in base 16, using

t-expansion [i] based on (25, 126, 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]

(126−99, 126, 156)-Net over F₁₆ — Digital

Digital (27, 126, 156)-net over F₁₆, using

net from sequence [i] based on digital (27, 155)-sequence over F₁₆, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₆ with g(F) = 27 and N(F) ≥ 156, using
  - a curve from the manYPoints database [i]

(126−99, 126, 1475)-Net in Base 16 — Upper bound on s

There is no (27, 126, 1476)-net in base 16, because

1 times m-reduction [i] would yield (27, 125, 1476)-net in base 16, but
- the generalized Rao bound for nets shows that 16^mÂ â‰¥ 3 291636 933133 847441 476027 004884 261871 624256 090737 108674 977555 750285 605799 938390 170975 440397 899102 574276 425481 655453 587928 706015 212093 890645 818412 364836 > 16¹²⁵ [i]