MinT - Best Known (171−61, 171, s)-Nets in Base 8

Best Known (171−61, 171, s)-Nets in Base 8

(171−61, 171, 382)-Net over F₈ — Constructive and digital

Digital (110, 171, 382)-net over F₈, using

(u,Â u+v)-construction [i] based on
1. digital (5, 35, 28)-net over F₈, using
  - net from sequence [i] based on digital (5, 27)-sequence over F₈, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈ with g(F) = 5 and N(F) ≥ 28, using
      - a function field by SÃ©mirat [i]
2. digital (75, 136, 354)-net over F₈, using
  - trace code for nets [i] based on digital (7, 68, 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]

(171−61, 171, 576)-Net in Base 8 — Constructive

(110, 171, 576)-net in base 8, using

t-expansion [i] based on (108, 171, 576)-net in base 8, using
- 1 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]

(171−61, 171, 1271)-Net over F₈ — Digital

Digital (110, 171, 1271)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(171−61, 171, 225509)-Net in Base 8 — Upper bound on s

There is no (110, 171, 225510)-net in base 8, because

1 times m-reduction [i] would yield (110, 170, 225510)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 3352 282003 981469 931932 385618 818603 686157 039913 583130 407201 668942 935467 478200 937173 666218 094880 967982 755658 589316 956157 442926 373039 446914 923952 632183 698000 > 8¹⁷⁰ [i]