MinT - Best Known (82, 82+39, s)-Nets in Base 8

Best Known (82, 82+39, s)-Nets in Base 8

(82, 82+39, 400)-Net over F₈ — Constructive and digital

Digital (82, 121, 400)-net over F₈, using

(u,Â u+v)-construction [i] based on
1. digital (10, 29, 46)-net over F₈, using
  - net from sequence [i] based on digital (10, 45)-sequence over F₈, using
    - Niederreiterâ€“Xing sequence constructionÂ III based on the algebraic function field F/F₈ with g(F)Â =Â 9, N(F)Â =Â 45, and 1 place with degree 2 [i] based on function field F/F₈ with g(F) = 9 and N(F) ≥ 45, using an explicitly constructive algebraic function field [i]
2. digital (53, 92, 354)-net over F₈, using
  - trace code for nets [i] based on digital (7, 46, 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]

(82, 82+39, 576)-Net in Base 8 — Constructive

(82, 121, 576)-net in base 8, using

t-expansion [i] based on (81, 121, 576)-net in base 8, using
- 5 times m-reduction [i] based on (81, 126, 576)-net in base 8, using
  - trace code for nets [i] based on (18, 63, 288)-net in base 64, using
    - base change [i] based on digital (9, 54, 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]

(82, 82+39, 1631)-Net over F₈ — Digital

Digital (82, 121, 1631)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(82, 82+39, 572578)-Net in Base 8 — Upper bound on s

There is no (82, 121, 572579)-net in base 8, because

1 times m-reduction [i] would yield (82, 120, 572579)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 2 348604 306114 287728 677348 173109 383036 743375 547504 327925 524428 664351 203141 795918 385295 757579 608318 687058 247856 > 8¹²⁰ [i]