MinT - Best Known (106, 165, s)-Nets in Base 8

Best Known (106, 165, s)-Nets in Base 8

(106, 165, 379)-Net over F₈ — Constructive and digital

Digital (106, 165, 379)-net over F₈, using

(u,Â u+v)-construction [i] based on
1. digital (4, 33, 25)-net over F₈, using
  - net from sequence [i] based on digital (4, 24)-sequence over F₈, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈ with g(F) = 4 and N(F) ≥ 25, using
      - a function field by SÃ©mirat [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]

(106, 165, 576)-Net in Base 8 — Constructive

(106, 165, 576)-net in base 8, using

t-expansion [i] based on (105, 165, 576)-net in base 8, using
- 3 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]

(106, 165, 1218)-Net over F₈ — Digital

Digital (106, 165, 1218)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(106, 165, 213361)-Net in Base 8 — Upper bound on s

There is no (106, 165, 213362)-net in base 8, because

1 times m-reduction [i] would yield (106, 164, 213362)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 12787 049817 997380 238082 976463 214595 548436 606210 637098 578167 186313 254658 447434 302062 567430 420744 062567 405155 233075 667490 482583 421063 469133 186361 116872 > 8¹⁶⁴ [i]