MinT - Best Known (56, 82, s)-Nets in Base 8

Best Known (56, 82, s)-Nets in Base 8

(56, 82, 378)-Net over F₈ — Constructive and digital

Digital (56, 82, 378)-net over F₈, using

(u,Â u+v)-construction [i] based on
1. digital (3, 16, 24)-net over F₈, using
  - net from sequence [i] based on digital (3, 23)-sequence over F₈, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈ with g(F) = 3 and N(F) ≥ 24, using
      - the Klein quartic over F₈ [i]
2. digital (40, 66, 354)-net over F₈, using
  - trace code for nets [i] based on digital (7, 33, 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]

(56, 82, 576)-Net in Base 8 — Constructive

(56, 82, 576)-net in base 8, using

trace code for nets [i] based on (15, 41, 288)-net in base 64, using
- 1 times m-reduction [i] based on (15, 42, 288)-net in base 64, using
  - base change [i] based on digital (9, 36, 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]

(56, 82, 1345)-Net over F₈ — Digital

Digital (56, 82, 1345)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(56, 82, 402451)-Net in Base 8 — Upper bound on s

There is no (56, 82, 402452)-net in base 8, because

the generalized Rao bound for nets shows that 8^mÂ â‰¥ 113 080467 745173 128727 569354 855912 204101 153991 889738 729161 237583 232749 138564 > 8⁸² [i]