MinT - Best Known (155−51, 155, s)-Nets in Base 8

Best Known (155−51, 155, s)-Nets in Base 8

(155−51, 155, 419)-Net over F₈ — Constructive and digital

Digital (104, 155, 419)-net over F₈, using

(u,Â u+v)-construction [i] based on
1. digital (14, 39, 65)-net over F₈, using
  - net from sequence [i] based on digital (14, 64)-sequence over F₈, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈ with g(F) = 14 and N(F) ≥ 65, using
      - the Suzuki function field over F₈ [i]
2. digital (65, 116, 354)-net over F₈, using
  - trace code for nets [i] based on digital (7, 58, 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]

(155−51, 155, 576)-Net in Base 8 — Constructive

(104, 155, 576)-net in base 8, using

11 times m-reduction [i] based on (104, 166, 576)-net in base 8, using
- trace code for nets [i] based on (21, 83, 288)-net in base 64, using
  - 1 times m-reduction [i] based on (21, 84, 288)-net in base 64, using
    - base change [i] based on digital (9, 72, 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]

(155−51, 155, 1779)-Net over F₈ — Digital

Digital (104, 155, 1779)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(155−51, 155, 531556)-Net in Base 8 — Upper bound on s

There is no (104, 155, 531557)-net in base 8, because

1 times m-reduction [i] would yield (104, 154, 531557)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 11 908838 664809 479315 353918 053329 855958 652651 801219 005088 866338 307059 844397 482130 301214 923246 802311 935572 422929 400818 822105 774056 349031 097232 > 8¹⁵⁴ [i]