MinT - Best Known (123−99, 123, s)-Nets in Base 16

Best Known (123−99, 123, s)-Nets in Base 16

(123−99, 123, 65)-Net over F₁₆ — Constructive and digital

Digital (24, 123, 65)-net over F₁₆, using

t-expansion [i] based on digital (6, 123, 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]

(123−99, 123, 129)-Net over F₁₆ — Digital

Digital (24, 123, 129)-net over F₁₆, using

t-expansion [i] based on digital (19, 123, 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]

(123−99, 123, 1241)-Net in Base 16 — Upper bound on s

There is no (24, 123, 1242)-net in base 16, because

1 times m-reduction [i] would yield (24, 122, 1242)-net in base 16, but
- the generalized Rao bound for nets shows that 16^mÂ â‰¥ 824 227112 080509 166493 698610 896621 088794 752618 020996 726073 946451 800047 160802 849680 935455 121904 730882 684173 262790 710761 846254 765654 574005 002041 020046 > 16¹²² [i]