MinT - Best Known (89−23, 89, s)-Nets in Base 8

Best Known (89−23, 89, s)-Nets in Base 8

(89−23, 89, 562)-Net over F₈ — Constructive and digital

Digital (66, 89, 562)-net over F₈, using

8¹ times duplication [i] based on digital (65, 88, 562)-net over F₈, using
- (u,Â u+v)-construction [i] based on
  1. digital (17, 28, 208)-net over F₈, using
    - trace code for nets [i] based on digital (3, 14, 104)-net over F₆₄, using
      - net from sequence [i] based on digital (3, 103)-sequence over F₆₄, using
        Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 3 and N(F) ≥ 104, using
        a function field by SÃ©mirat [i]
  2. digital (37, 60, 354)-net over F₈, using
    - trace code for nets [i] based on digital (7, 30, 177)-net over F₆₄, using
      - net from sequence [i] based on digital (7, 176)-sequence over F₆₄, using
        Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 7 and N(F) ≥ 177, using
        a function field by GarcÃa and Quoos [i]

(89−23, 89, 772)-Net in Base 8 — Constructive

(66, 89, 772)-net in base 8, using

8¹ times duplication [i] based on (65, 88, 772)-net in base 8, using
- (u,Â u+v)-construction [i] based on
  1. (15, 26, 258)-net in base 8, using
    - trace code for nets [i] based on (2, 13, 129)-net in base 64, using
      - 1 times m-reduction [i] based on (2, 14, 129)-net in base 64, using
        base change [i] based on digital (0, 12, 129)-net over F₁₂₈, using
        net from sequence [i] based on digital (0, 128)-sequence over F₁₂₈, using
        generalized Faure sequence [i]
        Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₂₈ with g(F) = 0 and N(F) ≥ 129, using
        the rational function field F₁₂₈(x) [i]
        Niederreiter sequence [i]
  2. (39, 62, 514)-net in base 8, using
    - trace code for nets [i] based on (8, 31, 257)-net in base 64, using
      - 1 times m-reduction [i] based on (8, 32, 257)-net in base 64, using
        base change [i] based on digital (0, 24, 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]

(89−23, 89, 5834)-Net over F₈ — Digital

Digital (66, 89, 5834)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(89−23, 89, large)-Net in Base 8 — Upper bound on s

There is no (66, 89, large)-net in base 8, because

21 times m-reduction [i] would yield (66, 68, large)-net in base 8, but
- mutually orthogonal hypercube bound [i]