MinT - Best Known (126−44, 126, s)-Nets in Base 8

Best Known (126−44, 126, s)-Nets in Base 8

(126−44, 126, 371)-Net over F₈ — Constructive and digital

Digital (82, 126, 371)-net over F₈, using

(u,Â u+v)-construction [i] based on
1. digital (2, 24, 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. 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]

(126−44, 126, 576)-Net in Base 8 — Constructive

(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]

(126−44, 126, 1090)-Net over F₈ — Digital

Digital (82, 126, 1090)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(126−44, 126, 192294)-Net in Base 8 — Upper bound on s

There is no (82, 126, 192295)-net in base 8, because

the generalized Rao bound for nets shows that 8^mÂ â‰¥ 615692 607703 507299 533624 576386 357753 613069 152817 479468 380829 975634 089167 815064 955070 870339 298673 470866 555624 290318 > 8¹²⁶ [i]