MinT - Best Known (128−105, 128, s)-Nets in Base 16

Best Known (128−105, 128, s)-Nets in Base 16

(128−105, 128, 65)-Net over F₁₆ — Constructive and digital

Digital (23, 128, 65)-net over F₁₆, using

t-expansion [i] based on digital (6, 128, 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]

(128−105, 128, 129)-Net over F₁₆ — Digital

Digital (23, 128, 129)-net over F₁₆, using

t-expansion [i] based on digital (19, 128, 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]

(128−105, 128, 1147)-Net in Base 16 — Upper bound on s

There is no (23, 128, 1148)-net in base 16, because

1 times m-reduction [i] would yield (23, 127, 1148)-net in base 16, but
- the generalized Rao bound for nets shows that 16^mÂ â‰¥ 844 274437 563420 709378 026283 741580 101119 319432 861299 606943 800593 966640 886599 417932 424367 413070 637826 410314 451548 354452 047346 970083 623792 908954 054090 555666 > 16¹²⁷ [i]