MinT - Best Known (172−66, 172, s)-Nets in Base 8

Best Known (172−66, 172, s)-Nets in Base 8

(172−66, 172, 354)-Net over F₈ — Constructive and digital

Digital (106, 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]

(172−66, 172, 432)-Net in Base 8 — Constructive

(106, 172, 432)-net in base 8, using

t-expansion [i] based on (104, 172, 432)-net in base 8, using
- trace code for nets [i] based on (18, 86, 216)-net in base 64, using
  - 5 times m-reduction [i] based on (18, 91, 216)-net in base 64, using
    - base change [i] based on digital (5, 78, 216)-net over F₁₂₈, using
      - net from sequence [i] based on digital (5, 215)-sequence over F₁₂₈, using
        Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₂₈ with g(F) = 5 and N(F) ≥ 216, using
        a function field by SÃ©mirat [i]

(172−66, 172, 877)-Net over F₈ — Digital

Digital (106, 172, 877)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(172−66, 172, 95758)-Net in Base 8 — Upper bound on s

There is no (106, 172, 95759)-net in base 8, because

the generalized Rao bound for nets shows that 8^mÂ â‰¥ 214573 534669 873081 786204 947396 051790 082088 688711 574719 183470 720536 028331 710892 048225 082718 735456 470329 115099 240022 726540 098292 490836 221509 535349 667681 849760 > 8¹⁷² [i]