MinT - Best Known (165−38, 165, s)-Nets in Base 8

Best Known (165−38, 165, s)-Nets in Base 8

(165−38, 165, 1091)-Net over F₈ — Constructive and digital

Digital (127, 165, 1091)-net over F₈, using

(u,Â u+v)-construction [i] based on
1. digital (14, 33, 65)-net over F₈, using
  - net from sequence [i] based on digital (14, 64)-sequence over F₈, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈ with g(F) = 14 and N(F) ≥ 65, using
      - the Suzuki function field over F₈ [i]
2. digital (94, 132, 1026)-net over F₈, using
  - trace code for nets [i] based on digital (28, 66, 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]

(165−38, 165, 1092)-Net in Base 8 — Constructive

(127, 165, 1092)-net in base 8, using

1 times m-reduction [i] based on (127, 166, 1092)-net in base 8, using
- (u,Â u+v)-construction [i] based on
  1. (35, 54, 516)-net in base 8, using
    - trace code for nets [i] based on (8, 27, 258)-net in base 64, using
      - 1 times m-reduction [i] based on (8, 28, 258)-net in base 64, using
        base change [i] based on digital (1, 21, 258)-net over F₂₅₆, using
        net from sequence [i] based on digital (1, 257)-sequence over F₂₅₆, using
        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]

(165−38, 165, 22308)-Net over F₈ — Digital

Digital (127, 165, 22308)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(165−38, 165, large)-Net in Base 8 — Upper bound on s

There is no (127, 165, large)-net in base 8, because

36 times m-reduction [i] would yield (127, 129, large)-net in base 8, but
- mutually orthogonal hypercube bound [i]