MinT - Best Known (112−37, 112, s)-Nets in Base 8

Best Known (112−37, 112, s)-Nets in Base 8

(112−37, 112, 382)-Net over F₈ — Constructive and digital

Digital (75, 112, 382)-net over F₈, using

8¹ times duplication [i] based on digital (74, 111, 382)-net over F₈, using
- (u,Â u+v)-construction [i] based on
  1. digital (5, 23, 28)-net over F₈, using
    - net from sequence [i] based on digital (5, 27)-sequence over F₈, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈ with g(F) = 5 and N(F) ≥ 28, using
        a function field by SÃ©mirat [i]
  2. digital (51, 88, 354)-net over F₈, using
    - trace code for nets [i] based on digital (7, 44, 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]

(112−37, 112, 576)-Net in Base 8 — Constructive

(75, 112, 576)-net in base 8, using

t-expansion [i] based on (73, 112, 576)-net in base 8, using
- trace code for nets [i] based on (17, 56, 288)-net in base 64, using
  - base change [i] based on digital (9, 48, 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]

(112−37, 112, 1334)-Net over F₈ — Digital

Digital (75, 112, 1334)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(112−37, 112, 400013)-Net in Base 8 — Upper bound on s

There is no (75, 112, 400014)-net in base 8, because

1 times m-reduction [i] would yield (75, 111, 400014)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 17498 205149 438225 077874 151152 700053 614933 135063 899601 328395 886897 646337 648590 417941 031725 453264 080240 > 8¹¹¹ [i]