MinT - Best Known (110−27, 110, s)-Nets in Base 8

Best Known (110−27, 110, s)-Nets in Base 8

(110−27, 110, 1026)-Net over F₈ — Constructive and digital

Digital (83, 110, 1026)-net over F₈, using

trace code for nets [i] based on digital (28, 55, 513)-net over F₆₄, using
- net from sequence [i] based on digital (28, 512)-sequence over F₆₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 28 and N(F) ≥ 513, using
    - the Hermitian function field over F₆₄ [i]

(110−27, 110, 1028)-Net in Base 8 — Constructive

(83, 110, 1028)-net in base 8, using

8² times duplication [i] based on (81, 108, 1028)-net in base 8, using
- base change [i] based on digital (54, 81, 1028)-net over F₁₆, using
  - 16¹ times duplication [i] based on digital (53, 80, 1028)-net over F₁₆, using
    - (u,Â u+v)-construction [i] based on
      1. digital (13, 26, 514)-net over F₁₆, using
        trace code for nets [i] based on digital (0, 13, 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]
      2. digital (27, 54, 514)-net over F₁₆, using
        trace code for nets [i] based on digital (0, 27, 257)-net over F₂₅₆, using
        net from sequence [i] based on digital (0, 256)-sequence over F₂₅₆ (see above)

(110−27, 110, 9989)-Net over F₈ — Digital

Digital (83, 110, 9989)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(110−27, 110, large)-Net in Base 8 — Upper bound on s

There is no (83, 110, large)-net in base 8, because

25 times m-reduction [i] would yield (83, 85, large)-net in base 8, but
- mutually orthogonal hypercube bound [i]