MinT - Best Known (107, 107+55, s)-Nets in Base 8

Best Known (107, 107+55, s)-Nets in Base 8

(107, 107+55, 402)-Net over F₈ — Constructive and digital

Digital (107, 162, 402)-net over F₈, using

(u,Â u+v)-construction [i] based on
1. digital (11, 38, 48)-net over F₈, using
  - net from sequence [i] based on digital (11, 47)-sequence over F₈, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈ with g(F) = 11 and N(F) ≥ 48, using
      - a function field by SÃ©mirat [i]
2. digital (69, 124, 354)-net over F₈, using
  - trace code for nets [i] based on digital (7, 62, 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]

(107, 107+55, 576)-Net in Base 8 — Constructive

(107, 162, 576)-net in base 8, using

t-expansion [i] based on (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]

(107, 107+55, 1561)-Net over F₈ — Digital

Digital (107, 162, 1561)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(107, 107+55, 378775)-Net in Base 8 — Upper bound on s

There is no (107, 162, 378776)-net in base 8, because

1 times m-reduction [i] would yield (107, 161, 378776)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 24 975395 154904 685219 615561 920088 309013 374730 944827 815533 163242 553521 061132 154386 782995 443391 658034 551599 899015 926704 848237 918270 435556 774263 722000 > 8¹⁶¹ [i]