MinT - Best Known (112−89, 112, s)-Nets in Base 16

Best Known (112−89, 112, s)-Nets in Base 16

(112−89, 112, 65)-Net over F₁₆ — Constructive and digital

Digital (23, 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]

(112−89, 112, 129)-Net over F₁₆ — Digital

Digital (23, 112, 129)-net over F₁₆, using

t-expansion [i] based on digital (19, 112, 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]

(112−89, 112, 1230)-Net in Base 16 — Upper bound on s

There is no (23, 112, 1231)-net in base 16, because

1 times m-reduction [i] would yield (23, 111, 1231)-net in base 16, but
- the generalized Rao bound for nets shows that 16^mÂ â‰¥ 46 618003 626812 685482 271918 215640 749378 884056 543792 805355 140521 912710 689767 637610 351691 271063 741502 235464 098836 800439 679489 260813 032686 > 16¹¹¹ [i]