MinT - Best Known (125−41, 125, s)-Nets in Base 8

Best Known (125−41, 125, s)-Nets in Base 8

(125−41, 125, 399)-Net over F₈ — Constructive and digital

Digital (84, 125, 399)-net over F₈, using

(u,Â u+v)-construction [i] based on
1. digital (9, 29, 45)-net over F₈, using
  - net from sequence [i] based on digital (9, 44)-sequence over F₈, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈ with g(F) = 9 and N(F) ≥ 45, using
      - an explicitly constructive algebraic function field [i]
2. digital (55, 96, 354)-net over F₈, using
  - trace code for nets [i] based on digital (7, 48, 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]

(125−41, 125, 576)-Net in Base 8 — Constructive

(84, 125, 576)-net in base 8, using

5 times m-reduction [i] based on (84, 130, 576)-net in base 8, using
- trace code for nets [i] based on (19, 65, 288)-net in base 64, using
  - 5 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]

(125−41, 125, 1516)-Net over F₈ — Digital

Digital (84, 125, 1516)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(125−41, 125, 471362)-Net in Base 8 — Upper bound on s

There is no (84, 125, 471363)-net in base 8, because

1 times m-reduction [i] would yield (84, 124, 471363)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 9619 968961 043670 582818 764898 842175 709985 838370 177163 318194 104534 241325 216009 000572 058338 360353 144219 541974 400784 > 8¹²⁴ [i]