MinT - Best Known (153−55, 153, s)-Nets in Base 8

Best Known (153−55, 153, s)-Nets in Base 8

(153−55, 153, 371)-Net over F₈ — Constructive and digital

Digital (98, 153, 371)-net over F₈, using

(u,Â u+v)-construction [i] based on
1. digital (2, 29, 17)-net over F₈, using
  - net from sequence [i] based on digital (2, 16)-sequence over F₈, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈ with g(F) = 2 and N(F) ≥ 17, using
      - a function field by SÃ©mirat [i]
2. digital (69, 124, 354)-net over F₈, using
  - trace code for nets [i] based on digital (7, 62, 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]

(153−55, 153, 576)-Net in Base 8 — Constructive

(98, 153, 576)-net in base 8, using

t-expansion [i] based on (97, 153, 576)-net in base 8, using
- 1 times m-reduction [i] based on (97, 154, 576)-net in base 8, using
  - trace code for nets [i] based on (20, 77, 288)-net in base 64, using
    - base change [i] based on digital (9, 66, 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]

(153−55, 153, 1111)-Net over F₈ — Digital

Digital (98, 153, 1111)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(153−55, 153, 189379)-Net in Base 8 — Upper bound on s

There is no (98, 153, 189380)-net in base 8, because

1 times m-reduction [i] would yield (98, 152, 189380)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 186092 554549 609009 818565 148965 042098 154835 621665 918320 148343 296681 936304 126640 983650 703575 691721 327052 529286 561013 875975 447764 210633 162408 > 8¹⁵² [i]