MinT - Best Known (112, 173, s)-Nets in Base 8

Best Known (112, 173, s)-Nets in Base 8

(112, 173, 388)-Net over F₈ — Constructive and digital

Digital (112, 173, 388)-net over F₈, using

(u,Â u+v)-construction [i] based on
1. digital (7, 37, 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 (75, 136, 354)-net over F₈, using
  - trace code for nets [i] based on digital (7, 68, 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]

(112, 173, 576)-Net in Base 8 — Constructive

(112, 173, 576)-net in base 8, using

8¹ times duplication [i] based on (111, 172, 576)-net in base 8, using
- t-expansion [i] based on (108, 172, 576)-net in base 8, using
  - trace code for nets [i] based on (22, 86, 288)-net in base 64, using
    - 5 times m-reduction [i] based on (22, 91, 288)-net in base 64, using
      - base change [i] based on digital (9, 78, 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]

(112, 173, 1361)-Net over F₈ — Digital

Digital (112, 173, 1361)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(112, 173, 259044)-Net in Base 8 — Upper bound on s

There is no (112, 173, 259045)-net in base 8, because

1 times m-reduction [i] would yield (112, 172, 259045)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 214527 184314 863794 069763 407090 019998 885018 389148 191551 897771 315385 505564 482507 836977 381027 351799 942830 945480 586463 880931 053943 110674 556142 667446 153115 312560 > 8¹⁷² [i]