MinT - Best Known (31−15, 31, s)-Nets in Base 16

Best Known (31−15, 31, s)-Nets in Base 16

(31−15, 31, 514)-Net over F₁₆ — Constructive and digital

Digital (16, 31, 514)-net over F₁₆, using

1 times m-reduction [i] based on digital (16, 32, 514)-net over F₁₆, using
- trace code for nets [i] based on digital (0, 16, 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]

(31−15, 31, 32605)-Net in Base 16 — Upper bound on s

There is no (16, 31, 32606)-net in base 16, because

1 times m-reduction [i] would yield (16, 30, 32606)-net in base 16, but
- the generalized Rao bound for nets shows that 16^mÂ â‰¥ 1 329272 861574 674664 175807 457762 390356 > 16³⁰ [i]