MinT - Best Known (112−41, 112, s)-Nets in Base 16

Best Known (112−41, 112, s)-Nets in Base 16

(112−41, 112, 583)-Net over F₁₆ — Constructive and digital

Digital (71, 112, 583)-net over F₁₆, using

(u,Â u+v)-construction [i] based on
1. digital (6, 26, 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]
2. digital (45, 86, 518)-net over F₁₆, using
  - trace code for nets [i] based on digital (2, 43, 259)-net over F₂₅₆, using
    - net from sequence [i] based on digital (2, 258)-sequence over F₂₅₆, using
      - Niederreiter sequence [i]

(112−41, 112, 2493)-Net over F₁₆ — Digital

Digital (71, 112, 2493)-net over F₁₆, using

Gilbertâ€“VarÅ¡amov bound [i]

(112−41, 112, 2667344)-Net in Base 16 — Upper bound on s

There is no (71, 112, 2667345)-net in base 16, because

1 times m-reduction [i] would yield (71, 111, 2667345)-net in base 16, but
- the generalized Rao bound for nets shows that 16^mÂ â‰¥ 45 427588 092482 011699 367809 816994 076663 606193 146982 048786 459664 306436 225975 005652 949365 107454 992927 865424 697465 318432 167802 615942 115376 > 16¹¹¹ [i]