MinT - Best Known (127−91, 127, s)-Nets in Base 16

Best Known (127−91, 127, s)-Nets in Base 16

(127−91, 127, 65)-Net over F₁₆ — Constructive and digital

Digital (36, 127, 65)-net over F₁₆, using

t-expansion [i] based on digital (6, 127, 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]

(127−91, 127, 104)-Net in Base 16 — Constructive

(36, 127, 104)-net in base 16, using

t-expansion [i] based on (35, 127, 104)-net in base 16, using
- 3 times m-reduction [i] based on (35, 130, 104)-net in base 16, using
  - base change [i] based on digital (9, 104, 104)-net over F₃₂, using
    - net from sequence [i] based on digital (9, 103)-sequence over F₃₂, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃₂ with g(F) = 9 and N(F) ≥ 104, using
        a function field by SÃ©mirat [i]

(127−91, 127, 193)-Net over F₁₆ — Digital

Digital (36, 127, 193)-net over F₁₆, using

t-expansion [i] based on digital (33, 127, 193)-net over F₁₆, using
- net from sequence [i] based on digital (33, 192)-sequence over F₁₆, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₆ with g(F) = 33 and N(F) ≥ 193, using
    - a curve from the manYPoints database [i]

(127−91, 127, 2739)-Net in Base 16 — Upper bound on s

There is no (36, 127, 2740)-net in base 16, because

1 times m-reduction [i] would yield (36, 126, 2740)-net in base 16, but
- the generalized Rao bound for nets shows that 16^mÂ â‰¥ 52 745012 205575 525158 486938 197265 328467 608711 564724 109606 247681 599192 648094 041738 286650 967827 238292 409055 310013 449351 341908 080177 628733 248177 269172 738876 > 16¹²⁶ [i]