MinT - Best Known (127−105, 127, s)-Nets in Base 16

Best Known (127−105, 127, s)-Nets in Base 16

(127−105, 127, 65)-Net over F₁₆ — Constructive and digital

Digital (22, 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]

(127−105, 127, 129)-Net over F₁₆ — Digital

Digital (22, 127, 129)-net over F₁₆, using

t-expansion [i] based on digital (19, 127, 129)-net over F₁₆, using
- net from sequence [i] based on digital (19, 128)-sequence over F₁₆, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₆ with g(F) = 19 and N(F) ≥ 129, using
    - a curve from the manYPoints database [i]

(127−105, 127, 1086)-Net in Base 16 — Upper bound on s

There is no (22, 127, 1087)-net in base 16, because

1 times m-reduction [i] would yield (22, 126, 1087)-net in base 16, but
- the generalized Rao bound for nets shows that 16^mÂ â‰¥ 52 995115 677767 447328 845308 489711 070209 691931 538208 153637 080062 343208 675113 268364 163072 935190 859905 588082 500700 189365 336063 623704 216753 353979 529344 625411 > 16¹²⁶ [i]