MinT - Best Known (127−44, 127, s)-Nets in Base 8

Best Known (127−44, 127, s)-Nets in Base 8

(127−44, 127, 378)-Net over F₈ — Constructive and digital

Digital (83, 127, 378)-net over F₈, using

(u,Â u+v)-construction [i] based on
1. digital (3, 25, 24)-net over F₈, using
  - net from sequence [i] based on digital (3, 23)-sequence over F₈, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈ with g(F) = 3 and N(F) ≥ 24, using
      - the Klein quartic over F₈ [i]
2. digital (58, 102, 354)-net over F₈, using
  - trace code for nets [i] based on digital (7, 51, 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]

(127−44, 127, 576)-Net in Base 8 — Constructive

(83, 127, 576)-net in base 8, using

8¹ times duplication [i] based on (82, 126, 576)-net in base 8, using
- t-expansion [i] based on (81, 126, 576)-net in base 8, using
  - trace code for nets [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]

(127−44, 127, 1143)-Net over F₈ — Digital

Digital (83, 127, 1143)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(127−44, 127, 211358)-Net in Base 8 — Upper bound on s

There is no (83, 127, 211359)-net in base 8, because

the generalized Rao bound for nets shows that 8^mÂ â‰¥ 4 925648 101600 787569 257873 487572 923106 706192 970011 524642 403037 545314 460965 940263 691163 192010 155994 580958 377395 945749 > 8¹²⁷ [i]