MinT - Best Known (163−63, 163, s)-Nets in Base 8

Best Known (163−63, 163, s)-Nets in Base 8

(163−63, 163, 354)-Net over F₈ — Constructive and digital

Digital (100, 163, 354)-net over F₈, using

t-expansion [i] based on digital (93, 163, 354)-net over F₈, using
- 9 times m-reduction [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]

(163−63, 163, 432)-Net in Base 8 — Constructive

(100, 163, 432)-net in base 8, using

3 times m-reduction [i] based on (100, 166, 432)-net in base 8, using
- trace code for nets [i] based on (17, 83, 216)-net in base 64, using
  - 1 times m-reduction [i] based on (17, 84, 216)-net in base 64, using
    - base change [i] based on digital (5, 72, 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]

(163−63, 163, 809)-Net over F₈ — Digital

Digital (100, 163, 809)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(163−63, 163, 92943)-Net in Base 8 — Upper bound on s

There is no (100, 163, 92944)-net in base 8, because

1 times m-reduction [i] would yield (100, 162, 92944)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 199 845051 337700 382236 174452 454120 530376 734023 630651 816244 356462 852837 765409 154270 162402 703058 663673 926541 008647 945285 815224 128843 943344 800358 275472 > 8¹⁶² [i]