MinT - Best Known (113−38, 113, s)-Nets in Base 8

Best Known (113−38, 113, s)-Nets in Base 8

(113−38, 113, 379)-Net over F₈ — Constructive and digital

Digital (75, 113, 379)-net over F₈, using

(u,Â u+v)-construction [i] based on
1. digital (4, 23, 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 (52, 90, 354)-net over F₈, using
  - trace code for nets [i] based on digital (7, 45, 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]

(113−38, 113, 576)-Net in Base 8 — Constructive

(75, 113, 576)-net in base 8, using

8¹ times duplication [i] based on (74, 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]

(113−38, 113, 1218)-Net over F₈ — Digital

Digital (75, 113, 1218)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(113−38, 113, 266138)-Net in Base 8 — Upper bound on s

There is no (75, 113, 266139)-net in base 8, because

the generalized Rao bound for nets shows that 8^mÂ â‰¥ 1 119936 035986 138559 393588 299865 528433 334273 613618 611180 569045 419732 427311 852937 735622 533905 725010 720128 > 8¹¹³ [i]