MinT - Best Known (123−90, 123, s)-Nets in Base 16

Best Known (123−90, 123, s)-Nets in Base 16

(123−90, 123, 65)-Net over F₁₆ — Constructive and digital

Digital (33, 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]

(123−90, 123, 98)-Net in Base 16 — Constructive

(33, 123, 98)-net in base 16, using

7 times m-reduction [i] based on (33, 130, 98)-net in base 16, using
- base change [i] based on digital (7, 104, 98)-net over F₃₂, using
  - net from sequence [i] based on digital (7, 97)-sequence over F₃₂, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃₂ with g(F) = 7 and N(F) ≥ 98, using
      - a function field by SÃ©mirat [i]

(123−90, 123, 193)-Net over F₁₆ — Digital

Digital (33, 123, 193)-net over F₁₆, using

net from sequence [i] based on digital (33, 192)-sequence over F₁₆, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₆ with g(F) = 33 and N(F) ≥ 193, using
  - a curve from the manYPoints database [i]

(123−90, 123, 2273)-Net in Base 16 — Upper bound on s

There is no (33, 123, 2274)-net in base 16, because

the generalized Rao bound for nets shows that 16^mÂ â‰¥ 13031 012452 292471 360919 719708 242996 455462 670154 465192 519538 616208 874816 606661 794320 621674 052058 101555 419282 010624 238183 875033 219585 768228 082805 750576 > 16¹²³ [i]