MinT - Best Known (172−67, 172, s)-Nets in Base 8

Best Known (172−67, 172, s)-Nets in Base 8

(172−67, 172, 354)-Net over F₈ — Constructive and digital

Digital (105, 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]

(172−67, 172, 432)-Net in Base 8 — Constructive

(105, 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]

(172−67, 172, 815)-Net over F₈ — Digital

Digital (105, 172, 815)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(172−67, 172, 89909)-Net in Base 8 — Upper bound on s

There is no (105, 172, 89910)-net in base 8, because

1 times m-reduction [i] would yield (105, 171, 89910)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 26823 461081 462531 953586 743210 800190 503750 832469 006399 848022 314618 693824 700226 257190 004195 864449 169104 326360 841321 324446 767193 830199 591605 927371 064202 597045 > 8¹⁷¹ [i]