MinT - Best Known (123−43, 123, s)-Nets in Base 8

Best Known (123−43, 123, s)-Nets in Base 8

(123−43, 123, 371)-Net over F₈ — Constructive and digital

Digital (80, 123, 371)-net over F₈, using

(u,Â u+v)-construction [i] based on
1. digital (2, 23, 17)-net over F₈, using
  - net from sequence [i] based on digital (2, 16)-sequence over F₈, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈ with g(F) = 2 and N(F) ≥ 17, using
      - a function field by SÃ©mirat [i]
2. digital (57, 100, 354)-net over F₈, using
  - trace code for nets [i] based on digital (7, 50, 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]

(123−43, 123, 576)-Net in Base 8 — Constructive

(80, 123, 576)-net in base 8, using

1 times m-reduction [i] based on (80, 124, 576)-net in base 8, using
- trace code for nets [i] based on (18, 62, 288)-net in base 64, using
  - 1 times m-reduction [i] based on (18, 63, 288)-net in base 64, using
    - base change [i] based on digital (9, 54, 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]

(123−43, 123, 1062)-Net over F₈ — Digital

Digital (80, 123, 1062)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(123−43, 123, 218721)-Net in Base 8 — Upper bound on s

There is no (80, 123, 218722)-net in base 8, because

1 times m-reduction [i] would yield (80, 122, 218722)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 150 313784 301971 347361 778394 299648 902167 382616 874913 558626 410037 695979 665246 385507 903249 639023 264223 797591 540840 > 8¹²² [i]