MinT - Best Known (57, 57+33, s)-Nets in Base 16

Best Known (57, 57+33, s)-Nets in Base 16

(57, 57+33, 581)-Net over F₁₆ — Constructive and digital

Digital (57, 90, 581)-net over F₁₆, using

(u,Â u+v)-construction [i] based on
1. digital (6, 22, 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 (35, 68, 516)-net over F₁₆, using
  - trace code for nets [i] based on digital (1, 34, 258)-net over F₂₅₆, using
    - net from sequence [i] based on digital (1, 257)-sequence over F₂₅₆, using
      - Niederreiter sequence [i]

(57, 57+33, 2093)-Net over F₁₆ — Digital

Digital (57, 90, 2093)-net over F₁₆, using

Gilbertâ€“VarÅ¡amov bound [i]

(57, 57+33, 2261326)-Net in Base 16 — Upper bound on s

There is no (57, 90, 2261327)-net in base 16, because

1 times m-reduction [i] would yield (57, 89, 2261327)-net in base 16, but
- the generalized Rao bound for nets shows that 16^mÂ â‰¥ 146784 692111 154131 572319 986268 354427 048519 699032 552745 814496 319527 526072 223277 607641 581996 430039 102104 106356 > 16⁸⁹ [i]