MinT - Best Known (88−35, 88, s)-Nets in Base 16

Best Known (88−35, 88, s)-Nets in Base 16

(88−35, 88, 538)-Net over F₁₆ — Constructive and digital

Digital (53, 88, 538)-net over F₁₆, using

(u,Â u+v)-construction [i] based on
1. digital (1, 18, 24)-net over F₁₆, using
  - net from sequence [i] based on digital (1, 23)-sequence over F₁₆, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₆ with g(F) = 1 and N(F) ≥ 24, using
      - a function field by SÃ©mirat [i]
2. digital (35, 70, 514)-net over F₁₆, using
  - trace code for nets [i] based on digital (0, 35, 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]

(88−35, 88, 1197)-Net over F₁₆ — Digital

Digital (53, 88, 1197)-net over F₁₆, using

Gilbertâ€“VarÅ¡amov bound [i]

(88−35, 88, 695199)-Net in Base 16 — Upper bound on s

There is no (53, 88, 695200)-net in base 16, because

1 times m-reduction [i] would yield (53, 87, 695200)-net in base 16, but
- the generalized Rao bound for nets shows that 16^mÂ â‰¥ 573 384385 613643 029763 569648 833199 261156 474383 942909 006613 181669 751864 678559 525506 246964 386670 842964 492251 > 16⁸⁷ [i]