MinT - Best Known (164−39, 164, s)-Nets in Base 8

Best Known (164−39, 164, s)-Nets in Base 8

(164−39, 164, 1074)-Net over F₈ — Constructive and digital

Digital (125, 164, 1074)-net over F₈, using

(u,Â u+v)-construction [i] based on
1. digital (11, 30, 48)-net over F₈, using
  - net from sequence [i] based on digital (11, 47)-sequence over F₈, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈ with g(F) = 11 and N(F) ≥ 48, using
      - a function field by SÃ©mirat [i]
2. digital (95, 134, 1026)-net over F₈, using
  - trace code for nets [i] based on digital (28, 67, 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]

(164−39, 164, 1090)-Net in Base 8 — Constructive

(125, 164, 1090)-net in base 8, using

(u,Â u+v)-construction [i] based on
1. (33, 52, 514)-net in base 8, using
  - base change [i] based on digital (20, 39, 514)-net over F₁₆, using
    - 1 times m-reduction [i] based on digital (20, 40, 514)-net over F₁₆, using
      - trace code for nets [i] based on digital (0, 20, 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. (73, 112, 576)-net in base 8, using
  - trace code for nets [i] based on (17, 56, 288)-net in base 64, using
    - base change [i] based on digital (9, 48, 288)-net over F₁₂₈, using
      - net from sequence [i] based on digital (9, 287)-sequence over F₁₂₈, using
        Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₂₈ with g(F) = 9 and N(F) ≥ 288, using
        a function field by SÃ©mirat [i]

(164−39, 164, 16972)-Net over F₈ — Digital

Digital (125, 164, 16972)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(164−39, 164, large)-Net in Base 8 — Upper bound on s

There is no (125, 164, large)-net in base 8, because

37 times m-reduction [i] would yield (125, 127, large)-net in base 8, but
- mutually orthogonal hypercube bound [i]