MinT - Best Known (127−101, 127, s)-Nets in Base 16

Best Known (127−101, 127, s)-Nets in Base 16

(127−101, 127, 65)-Net over F₁₆ — Constructive and digital

Digital (26, 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]

(127−101, 127, 66)-Net in Base 16 — Constructive

(26, 127, 66)-net in base 16, using

t-expansion [i] based on (25, 127, 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]

(127−101, 127, 150)-Net over F₁₆ — Digital

Digital (26, 127, 150)-net over F₁₆, using

net from sequence [i] based on digital (26, 149)-sequence over F₁₆, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₆ with g(F) = 26 and N(F) ≥ 150, using
  - a curve from the manYPoints database [i]

(127−101, 127, 1378)-Net in Base 16 — Upper bound on s

There is no (26, 127, 1379)-net in base 16, because

1 times m-reduction [i] would yield (26, 126, 1379)-net in base 16, but
- the generalized Rao bound for nets shows that 16^mÂ â‰¥ 53 893230 875674 985912 265814 814432 977592 176446 016033 388615 629863 626682 302919 757670 234016 504898 415595 851888 635896 254610 012700 037597 999094 459364 921943 524876 > 16¹²⁶ [i]