MinT - Best Known (169−57, 169, s)-Nets in Base 8

Best Known (169−57, 169, s)-Nets in Base 8

(169−57, 169, 513)-Net over F₈ — Constructive and digital

Digital (112, 169, 513)-net over F₈, using

base reduction for projective spaces (embedding PG(84,64) in PG(168,8)) for nets [i] based on digital (28, 85, 513)-net over F₆₄, using
- net from sequence [i] based on digital (28, 512)-sequence over F₆₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 28 and N(F) ≥ 513, using
    - the Hermitian function field over F₆₄ [i]

(169−57, 169, 576)-Net in Base 8 — Constructive

(112, 169, 576)-net in base 8, using

t-expansion [i] based on (108, 169, 576)-net in base 8, using
- 3 times m-reduction [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]

(169−57, 169, 1676)-Net over F₈ — Digital

Digital (112, 169, 1676)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(169−57, 169, 423086)-Net in Base 8 — Upper bound on s

There is no (112, 169, 423087)-net in base 8, because

1 times m-reduction [i] would yield (112, 168, 423087)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 52 376724 341475 344488 368497 569712 003933 078890 756490 558890 344305 840441 589950 563495 697863 973809 763145 276205 147435 275225 964195 401782 915027 420015 876050 802054 > 8¹⁶⁸ [i]