MinT - Best Known (48, 48+31, s)-Nets in Base 16

Best Known (48, 48+31, s)-Nets in Base 16

(48, 48+31, 547)-Net over F₁₆ — Constructive and digital

Digital (48, 79, 547)-net over F₁₆, using

(u,Â u+v)-construction [i] based on
1. digital (2, 17, 33)-net over F₁₆, using
  - net from sequence [i] based on digital (2, 32)-sequence over F₁₆, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₆ with g(F) = 2 and N(F) ≥ 33, using
      - a function field by SÃ©mirat [i]
2. digital (31, 62, 514)-net over F₁₆, using
  - trace code for nets [i] based on digital (0, 31, 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]

(48, 48+31, 1205)-Net over F₁₆ — Digital

Digital (48, 79, 1205)-net over F₁₆, using

Gilbertâ€“VarÅ¡amov bound [i]

(48, 48+31, 781798)-Net in Base 16 — Upper bound on s

There is no (48, 79, 781799)-net in base 16, because

1 times m-reduction [i] would yield (48, 78, 781799)-net in base 16, but
- the generalized Rao bound for nets shows that 16^mÂ â‰¥ 8343 775183 109001 215990 151231 583046 971674 667746 794309 631869 229432 104094 710222 128872 865279 079776 > 16⁷⁸ [i]