MinT - Best Known (119−91, 119, s)-Nets in Base 16

Best Known (119−91, 119, s)-Nets in Base 16

(119−91, 119, 65)-Net over F₁₆ — Constructive and digital

Digital (28, 119, 65)-net over F₁₆, using

t-expansion [i] based on digital (6, 119, 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]

(119−91, 119, 66)-Net in Base 16 — Constructive

(28, 119, 66)-net in base 16, using

t-expansion [i] based on (25, 119, 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]

(119−91, 119, 156)-Net over F₁₆ — Digital

Digital (28, 119, 156)-net over F₁₆, using

t-expansion [i] based on digital (27, 119, 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]

(119−91, 119, 1663)-Net in Base 16 — Upper bound on s

There is no (28, 119, 1664)-net in base 16, because

1 times m-reduction [i] would yield (28, 118, 1664)-net in base 16, but
- the generalized Rao bound for nets shows that 16^mÂ â‰¥ 12262 343635 966167 976135 470111 958680 631810 770796 294966 411434 497193 380353 545168 975929 666222 968971 304196 064742 162712 557616 818857 524952 317722 955701 > 16¹¹⁸ [i]