MinT - Best Known (123−38, 123, s)-Nets in Base 8

Best Known (123−38, 123, s)-Nets in Base 8

(123−38, 123, 419)-Net over F₈ — Constructive and digital

Digital (85, 123, 419)-net over F₈, using

(u,Â u+v)-construction [i] based on
1. digital (14, 33, 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 (52, 90, 354)-net over F₈, using
  - trace code for nets [i] based on digital (7, 45, 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]

(123−38, 123, 576)-Net in Base 8 — Constructive

(85, 123, 576)-net in base 8, using

9 times m-reduction [i] based on (85, 132, 576)-net in base 8, using
- trace code for nets [i] based on (19, 66, 288)-net in base 64, using
  - 4 times m-reduction [i] based on (19, 70, 288)-net in base 64, using
    - base change [i] based on digital (9, 60, 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]

(123−38, 123, 2122)-Net over F₈ — Digital

Digital (85, 123, 2122)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(123−38, 123, 795115)-Net in Base 8 — Upper bound on s

There is no (85, 123, 795116)-net in base 8, because

the generalized Rao bound for nets shows that 8^mÂ â‰¥ 1202 455656 560078 776604 418712 416340 521945 869952 914974 009060 484831 780475 139512 173653 062379 600994 916875 698045 110330 > 8¹²³ [i]