MinT - Best Known (73, 73+47, s)-Nets in Base 16

Best Known (73, 73+47, s)-Nets in Base 16

(73, 73+47, 552)-Net over F₁₆ — Constructive and digital

Digital (73, 120, 552)-net over F₁₆, using

(u,Â u+v)-construction [i] based on
1. digital (3, 26, 38)-net over F₁₆, using
  - net from sequence [i] based on digital (3, 37)-sequence over F₁₆, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₆ with g(F) = 3 and N(F) ≥ 38, using
      - a function field by SÃ©mirat [i]
2. digital (47, 94, 514)-net over F₁₆, using
  - trace code for nets [i] based on digital (0, 47, 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]

(73, 73+47, 1684)-Net over F₁₆ — Digital

Digital (73, 120, 1684)-net over F₁₆, using

Gilbertâ€“VarÅ¡amov bound [i]

(73, 73+47, 1067510)-Net in Base 16 — Upper bound on s

There is no (73, 120, 1067511)-net in base 16, because

1 times m-reduction [i] would yield (73, 119, 1067511)-net in base 16, but
- the generalized Rao bound for nets shows that 16^mÂ â‰¥ 195112 932969 269247 039580 190097 257559 231998 905334 759072 356755 204617 184619 569411 491074 142102 464628 739128 795128 988111 146770 061778 679194 896068 106096 > 16¹¹⁹ [i]