MinT - Best Known (128−37, 128, s)-Nets in Base 8

Best Known (128−37, 128, s)-Nets in Base 8

(128−37, 128, 514)-Net over F₈ — Constructive and digital

Digital (91, 128, 514)-net over F₈, using

8² times duplication [i] based on digital (89, 126, 514)-net over F₈, using
- (u,Â u+v)-construction [i] based on
  1. digital (20, 38, 160)-net over F₈, using
    - trace code for nets [i] based on digital (1, 19, 80)-net over F₆₄, using
      - net from sequence [i] based on digital (1, 79)-sequence over F₆₄, using
        Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 1 and N(F) ≥ 80, 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]

(128−37, 128, 593)-Net in Base 8 — Constructive

(91, 128, 593)-net in base 8, using

(u,Â u+v)-construction [i] based on
1. digital (2, 20, 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. (71, 108, 576)-net in base 8, using
  - trace code for nets [i] based on (17, 54, 288)-net in base 64, using
    - 2 times m-reduction [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]

(128−37, 128, 3334)-Net over F₈ — Digital

Digital (91, 128, 3334)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(128−37, 128, 2539988)-Net in Base 8 — Upper bound on s

There is no (91, 128, 2539989)-net in base 8, because

1 times m-reduction [i] would yield (91, 127, 2539989)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 4 925278 797374 491835 816303 372811 340083 403500 007471 558185 861808 176957 222701 914027 924791 426989 577203 496654 819255 866390 > 8¹²⁷ [i]