MinT - Best Known (126−98, 126, s)-Nets in Base 16

Best Known (126−98, 126, s)-Nets in Base 16

(126−98, 126, 65)-Net over F₁₆ — Constructive and digital

Digital (28, 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]

(126−98, 126, 66)-Net in Base 16 — Constructive

(28, 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]

(126−98, 126, 156)-Net over F₁₆ — Digital

Digital (28, 126, 156)-net over F₁₆, using

t-expansion [i] based on 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]

(126−98, 126, 1563)-Net in Base 16 — Upper bound on s

There is no (28, 126, 1564)-net in base 16, because

the generalized Rao bound for nets shows that 16^mÂ â‰¥ 53 485549 441796 027472 202328 664761 997338 628932 316665 670714 493712 033353 262571 802981 956524 628463 279640 409644 147094 017980 210166 719732 184430 861346 990541 774416 > 16¹²⁶ [i]