MinT - Best Known (51, 51+30, s)-Nets in Base 16

Best Known (51, 51+30, s)-Nets in Base 16

(51, 51+30, 579)-Net over F₁₆ — Constructive and digital

Digital (51, 81, 579)-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 (30, 60, 514)-net over F₁₆, using
  - trace code for nets [i] based on digital (0, 30, 257)-net over F₂₅₆, using
    - net from sequence [i] based on digital (0, 256)-sequence over F₂₅₆, using
      - generalized Faure sequence [i]
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₅₆ with g(F) = 0 and N(F) ≥ 257, using
        the rational function field F₂₅₆(x) [i]
      - Niederreiter sequence [i]

(51, 51+30, 1810)-Net over F₁₆ — Digital

Digital (51, 81, 1810)-net over F₁₆, using

Gilbertâ€“VarÅ¡amov bound [i]

(51, 51+30, 1361196)-Net in Base 16 — Upper bound on s

There is no (51, 81, 1361197)-net in base 16, because

the generalized Rao bound for nets shows that 16^mÂ â‰¥ 34 176084 902339 668387 845255 463495 223154 236039 564310 113670 873480 381298 274903 789190 580587 998332 804576 > 16⁸¹ [i]