MinT - Best Known (164−59, 164, s)-Nets in Base 8

Best Known (164−59, 164, s)-Nets in Base 8

(164−59, 164, 378)-Net over F₈ — Constructive and digital

Digital (105, 164, 378)-net over F₈, using

(u,Â u+v)-construction [i] based on
1. digital (3, 32, 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 (73, 132, 354)-net over F₈, using
  - trace code for nets [i] based on digital (7, 66, 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]

(164−59, 164, 576)-Net in Base 8 — Constructive

(105, 164, 576)-net in base 8, using

4 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]

(164−59, 164, 1174)-Net over F₈ — Digital

Digital (105, 164, 1174)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(164−59, 164, 198597)-Net in Base 8 — Upper bound on s

There is no (105, 164, 198598)-net in base 8, because

1 times m-reduction [i] would yield (105, 163, 198598)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 1598 549438 427430 028361 553693 198014 583144 082278 630296 278127 866984 259700 439048 375754 665977 946793 157871 565525 679596 346469 202596 233086 661205 121312 588376 > 8¹⁶³ [i]