MinT - Best Known (172−64, 172, s)-Nets in Base 8

Best Known (172−64, 172, s)-Nets in Base 8

(172−64, 172, 354)-Net over F₈ — Constructive and digital

Digital (108, 172, 354)-net over F₈, using

t-expansion [i] based on digital (93, 172, 354)-net over F₈, using
- trace code for nets [i] based on digital (7, 86, 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]

(172−64, 172, 576)-Net in Base 8 — Constructive

(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]

(172−64, 172, 1024)-Net over F₈ — Digital

Digital (108, 172, 1024)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(172−64, 172, 130564)-Net in Base 8 — Upper bound on s

There is no (108, 172, 130565)-net in base 8, because

the generalized Rao bound for nets shows that 8^mÂ â‰¥ 214564 773687 092927 535392 590482 007138 257788 374855 420443 411115 119706 595922 609644 438234 635754 647737 288153 912501 670998 611999 595943 693611 650497 122240 754748 757904 > 8¹⁷² [i]