MinT - Best Known (162−57, 162, s)-Nets in Base 8

Best Known (162−57, 162, s)-Nets in Base 8

(162−57, 162, 382)-Net over F₈ — Constructive and digital

Digital (105, 162, 382)-net over F₈, using

8¹ times duplication [i] based on digital (104, 161, 382)-net over F₈, using
- (u,Â u+v)-construction [i] based on
  1. digital (5, 33, 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 (71, 128, 354)-net over F₈, using
    - trace code for nets [i] based on digital (7, 64, 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]

(162−57, 162, 576)-Net in Base 8 — Constructive

(105, 162, 576)-net in base 8, using

6 times m-reduction [i] based on (105, 168, 576)-net in base 8, using
- trace code for nets [i] based on (21, 84, 288)-net in base 64, using
  - base change [i] based on digital (9, 72, 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]

(162−57, 162, 1299)-Net over F₈ — Digital

Digital (105, 162, 1299)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(162−57, 162, 251561)-Net in Base 8 — Upper bound on s

There is no (105, 162, 251562)-net in base 8, because

1 times m-reduction [i] would yield (105, 161, 251562)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 24 975332 009827 531778 122618 197559 978538 581053 881183 749170 943209 094001 721202 148868 355220 637971 932475 337214 526177 658153 976441 234754 206335 827621 740072 > 8¹⁶¹ [i]