MinT - Best Known (127−107, 127, s)-Nets in Base 16

Best Known (127−107, 127, s)-Nets in Base 16

(127−107, 127, 65)-Net over F₁₆ — Constructive and digital

Digital (20, 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]

(127−107, 127, 129)-Net over F₁₆ — Digital

Digital (20, 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]

(127−107, 127, 971)-Net in Base 16 — Upper bound on s

There is no (20, 127, 972)-net in base 16, because

1 times m-reduction [i] would yield (20, 126, 972)-net in base 16, but
- the generalized Rao bound for nets shows that 16^mÂ â‰¥ 53 986635 581083 206690 632271 378548 166925 858238 842456 743712 350014 276229 084356 285169 127242 118742 970592 111025 240143 104338 245856 272790 166921 385853 627281 887016 > 16¹²⁶ [i]