MinT - Best Known (112−87, 112, s)-Nets in Base 16

Best Known (112−87, 112, s)-Nets in Base 16

(112−87, 112, 65)-Net over F₁₆ — Constructive and digital

Digital (25, 112, 65)-net over F₁₆, using

t-expansion [i] based on digital (6, 112, 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]

(112−87, 112, 66)-Net in Base 16 — Constructive

(25, 112, 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]

(112−87, 112, 144)-Net over F₁₆ — Digital

Digital (25, 112, 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]

(112−87, 112, 1420)-Net in Base 16 — Upper bound on s

There is no (25, 112, 1421)-net in base 16, because

1 times m-reduction [i] would yield (25, 111, 1421)-net in base 16, but
- the generalized Rao bound for nets shows that 16^mÂ â‰¥ 45 931899 069156 724109 879004 973563 506033 523583 606873 330138 855682 677069 706272 246792 485008 985046 590890 043896 519695 294343 182596 384421 942096 > 16¹¹¹ [i]