MinT - Best Known (173−60, 173, s)-Nets in Base 8

Best Known (173−60, 173, s)-Nets in Base 8

(173−60, 173, 399)-Net over F₈ — Constructive and digital

Digital (113, 173, 399)-net over F₈, using

(u,Â u+v)-construction [i] based on
1. digital (9, 39, 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 (74, 134, 354)-net over F₈, using
  - trace code for nets [i] based on digital (7, 67, 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]

(173−60, 173, 576)-Net in Base 8 — Constructive

(113, 173, 576)-net in base 8, using

8¹ times duplication [i] based on (112, 172, 576)-net in base 8, using
- t-expansion [i] based on (108, 172, 576)-net in base 8, using
  - trace code for nets [i] based on (22, 86, 288)-net in base 64, using
    - 5 times m-reduction [i] based on (22, 91, 288)-net in base 64, using
      - base change [i] based on digital (9, 78, 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]

(173−60, 173, 1479)-Net over F₈ — Digital

Digital (113, 173, 1479)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(173−60, 173, 277638)-Net in Base 8 — Upper bound on s

There is no (113, 173, 277639)-net in base 8, because

the generalized Rao bound for nets shows that 8^mÂ â‰¥ 1 716235 116503 178673 901408 075202 445080 183611 474109 203617 511399 057496 738202 342886 836301 596464 809146 826732 639564 928216 250464 582918 789217 544772 914978 157063 493296 > 8¹⁷³ [i]