MinT - Best Known (53, 53+30, s)-Nets in Base 16

Best Known (53, 53+30, s)-Nets in Base 16

(53, 53+30, 581)-Net over F₁₆ — Constructive and digital

Digital (53, 83, 581)-net over F₁₆, using

(u,Â u+v)-construction [i] based on
1. digital (6, 21, 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 (32, 62, 516)-net over F₁₆, using
  - trace code for nets [i] based on digital (1, 31, 258)-net over F₂₅₆, using
    - net from sequence [i] based on digital (1, 257)-sequence over F₂₅₆, using
      - Niederreiter sequence [i]

(53, 53+30, 2189)-Net over F₁₆ — Digital

Digital (53, 83, 2189)-net over F₁₆, using

Gilbertâ€“VarÅ¡amov bound [i]

(53, 53+30, 1970020)-Net in Base 16 — Upper bound on s

There is no (53, 83, 1970021)-net in base 16, because

the generalized Rao bound for nets shows that 16^mÂ â‰¥ 8749 002969 085082 672385 125688 497010 687031 081813 118603 494481 491710 893009 489836 476557 611480 166989 578976 > 16⁸³ [i]