MinT - Best Known (124−36, 124, s)-Nets in Base 8

Best Known (124−36, 124, s)-Nets in Base 8

(124−36, 124, 514)-Net over F₈ — Constructive and digital

Digital (88, 124, 514)-net over F₈, using

(u,Â u+v)-construction [i] based on
1. digital (20, 38, 160)-net over F₈, using
  - trace code for nets [i] based on digital (1, 19, 80)-net over F₆₄, using
    - net from sequence [i] based on digital (1, 79)-sequence over F₆₄, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 1 and N(F) ≥ 80, using
        a function field by SÃ©mirat [i]
2. digital (50, 86, 354)-net over F₈, using
  - trace code for nets [i] based on digital (7, 43, 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]

(124−36, 124, 585)-Net in Base 8 — Constructive

(88, 124, 585)-net in base 8, using

(u,Â u+v)-construction [i] based on
1. digital (0, 18, 9)-net over F₈, using
  - net from sequence [i] based on digital (0, 8)-sequence over F₈, using
    - generalized Faure sequence [i]
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈ with g(F) = 0 and N(F) ≥ 9, using
      - the rational function field F₈(x) [i]
    - Niederreiter sequence [i]
2. (70, 106, 576)-net in base 8, using
  - trace code for nets [i] based on (17, 53, 288)-net in base 64, using
    - 3 times m-reduction [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]

(124−36, 124, 3163)-Net over F₈ — Digital

Digital (88, 124, 3163)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(124−36, 124, 1796039)-Net in Base 8 — Upper bound on s

There is no (88, 124, 1796040)-net in base 8, because

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]