MinT - Best Known (171−60, 171, s)-Nets in Base 8

Best Known (171−60, 171, s)-Nets in Base 8

(171−60, 171, 388)-Net over F₈ — Constructive and digital

Digital (111, 171, 388)-net over F₈, using

(u,Â u+v)-construction [i] based on
1. digital (7, 37, 34)-net over F₈, using
  - net from sequence [i] based on digital (7, 33)-sequence over F₈, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈ with g(F) = 7 and N(F) ≥ 34, using
      - a function field by SÃ©mirat [i]
2. digital (74, 134, 354)-net over F₈, using
  - trace code for nets [i] based on digital (7, 67, 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]

(171−60, 171, 576)-Net in Base 8 — Constructive

(111, 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]

(171−60, 171, 1381)-Net over F₈ — Digital

Digital (111, 171, 1381)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(171−60, 171, 241696)-Net in Base 8 — Upper bound on s

There is no (111, 171, 241697)-net in base 8, because

the generalized Rao bound for nets shows that 8^mÂ â‰¥ 26818 316658 338416 074205 227210 294066 339155 849926 855447 268621 426395 670455 061485 952735 264863 610361 528215 485711 463865 168994 777305 925591 559352 731027 488927 481072 > 8¹⁷¹ [i]