MinT - Best Known (82−25, 82, s)-Nets in Base 8

Best Known (82−25, 82, s)-Nets in Base 8

(82−25, 82, 382)-Net over F₈ — Constructive and digital

Digital (57, 82, 382)-net over F₈, using

8¹ times duplication [i] based on digital (56, 81, 382)-net over F₈, using
- (u,Â u+v)-construction [i] based on
  1. digital (5, 17, 28)-net over F₈, using
    - net from sequence [i] based on digital (5, 27)-sequence over F₈, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈ with g(F) = 5 and N(F) ≥ 28, using
        a function field by SÃ©mirat [i]
  2. digital (39, 64, 354)-net over F₈, using
    - trace code for nets [i] based on digital (7, 32, 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]

(82−25, 82, 576)-Net in Base 8 — Constructive

(57, 82, 576)-net in base 8, using

2 times m-reduction [i] based on (57, 84, 576)-net in base 8, using
- trace code for nets [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]

(82−25, 82, 1717)-Net over F₈ — Digital

Digital (57, 82, 1717)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(82−25, 82, 942144)-Net in Base 8 — Upper bound on s

There is no (57, 82, 942145)-net in base 8, because

1 times m-reduction [i] would yield (57, 81, 942145)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 14 134888 724116 517494 837960 502571 413070 342755 295177 056810 670026 364921 167312 > 8⁸¹ [i]