MinT - Best Known (125−37, 125, s)-Nets in Base 8

Best Known (125−37, 125, s)-Nets in Base 8

(125−37, 125, 484)-Net over F₈ — Constructive and digital

Digital (88, 125, 484)-net over F₈, using

8¹ times duplication [i] based on digital (87, 124, 484)-net over F₈, using
- (u,Â u+v)-construction [i] based on
  1. digital (18, 36, 130)-net over F₈, using
    - trace code for nets [i] based on digital (0, 18, 65)-net over F₆₄, using
      - net from sequence [i] based on digital (0, 64)-sequence over F₆₄, using
        generalized Faure sequence [i]
        Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 0 and N(F) ≥ 65, using
        the rational function field F₆₄(x) [i]
        Niederreiter sequence [i]
  2. digital (51, 88, 354)-net over F₈, using
    - trace code for nets [i] based on digital (7, 44, 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]

(125−37, 125, 576)-Net in Base 8 — Constructive

(88, 125, 576)-net in base 8, using

13 times m-reduction [i] based on (88, 138, 576)-net in base 8, using
- trace code for nets [i] based on (19, 69, 288)-net in base 64, using
  - 1 times m-reduction [i] based on (19, 70, 288)-net in base 64, using
    - base change [i] based on digital (9, 60, 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]

(125−37, 125, 2806)-Net over F₈ — Digital

Digital (88, 125, 2806)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(125−37, 125, 1796039)-Net in Base 8 — Upper bound on s

There is no (88, 125, 1796040)-net in base 8, because

1 times m-reduction [i] would yield (88, 124, 1796040)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 9619 665550 978339 940072 686859 339034 462735 759174 120708 453631 268874 673258 948561 270463 320342 044343 227696 075207 063556 > 8¹²⁴ [i]