MinT - Best Known (86, 86+44, s)-Nets in Base 16

Best Known (86, 86+44, s)-Nets in Base 16

(86, 86+44, 617)-Net over F₁₆ — Constructive and digital

Digital (86, 130, 617)-net over F₁₆, using

(u,Â u+v)-construction [i] based on
1. digital (20, 42, 103)-net over F₁₆, using
  - (u,Â u+v)-construction [i] based on
    1. digital (3, 14, 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 (6, 28, 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 (44, 88, 514)-net over F₁₆, using
  - trace code for nets [i] based on digital (0, 44, 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]

(86, 86+44, 664)-Net in Base 16 — Constructive

(86, 130, 664)-net in base 16, using

16¹ times duplication [i] based on (85, 129, 664)-net in base 16, using
- (u,Â u+v)-construction [i] based on
  1. (19, 41, 150)-net in base 16, using
    - 1 times m-reduction [i] based on (19, 42, 150)-net in base 16, using
      - base change [i] based on digital (1, 24, 150)-net over F₁₂₈, using
        net from sequence [i] based on digital (1, 149)-sequence over F₁₂₈, using
        Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₂₈ with g(F) = 1 and N(F) ≥ 150, using
        a function field by SÃ©mirat [i]
  2. digital (44, 88, 514)-net over F₁₆, using
    - trace code for nets [i] based on digital (0, 44, 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]

(86, 86+44, 4939)-Net over F₁₆ — Digital

Digital (86, 130, 4939)-net over F₁₆, using

Gilbertâ€“VarÅ¡amov bound [i]

(86, 86+44, 7870739)-Net in Base 16 — Upper bound on s

There is no (86, 130, 7870740)-net in base 16, because

the generalized Rao bound for nets shows that 16^mÂ â‰¥ 3 432401 563595 435172 873316 527015 490784 091406 215146 166221 367616 339819 525174 094081 453012 682883 292679 807512 051581 626848 896742 526374 931812 952934 422639 811298 320576 > 16¹³⁰ [i]