MinT - Best Known (161−56, 161, s)-Nets in Base 8

Best Known (161−56, 161, s)-Nets in Base 8

(161−56, 161, 388)-Net over F₈ — Constructive and digital

Digital (105, 161, 388)-net over F₈, using

(u,Â u+v)-construction [i] based on
1. digital (7, 35, 34)-net over F₈, using
  - net from sequence [i] based on digital (7, 33)-sequence over F₈, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈ with g(F) = 7 and N(F) ≥ 34, using
      - a function field by SÃ©mirat [i]
2. digital (70, 126, 354)-net over F₈, using
  - trace code for nets [i] based on digital (7, 63, 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]

(161−56, 161, 576)-Net in Base 8 — Constructive

(105, 161, 576)-net in base 8, using

7 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]

(161−56, 161, 1369)-Net over F₈ — Digital

Digital (105, 161, 1369)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(161−56, 161, 251561)-Net in Base 8 — Upper bound on s

There is no (105, 161, 251562)-net in base 8, because

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]