MinT - Best Known (127−49, 127, s)-Nets in Base 8

Best Known (127−49, 127, s)-Nets in Base 8

(127−49, 127, 354)-Net over F₈ — Constructive and digital

Digital (78, 127, 354)-net over F₈, using

15 times m-reduction [i] based on digital (78, 142, 354)-net over F₈, using
- trace code for nets [i] based on digital (7, 71, 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]

(127−49, 127, 432)-Net in Base 8 — Constructive

(78, 127, 432)-net in base 8, using

8¹ times duplication [i] based on (77, 126, 432)-net in base 8, using
- trace code for nets [i] based on (14, 63, 216)-net in base 64, using
  - base change [i] based on digital (5, 54, 216)-net over F₁₂₈, using
    - net from sequence [i] based on digital (5, 215)-sequence over F₁₂₈, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₂₈ with g(F) = 5 and N(F) ≥ 216, using
        a function field by SÃ©mirat [i]

(127−49, 127, 666)-Net over F₈ — Digital

Digital (78, 127, 666)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(127−49, 127, 77161)-Net in Base 8 — Upper bound on s

There is no (78, 127, 77162)-net in base 8, because

1 times m-reduction [i] would yield (78, 126, 77162)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 615668 522757 653195 370661 389167 286711 632176 101819 797001 348870 953736 389249 066159 673885 126055 583340 425680 136347 043318 > 8¹²⁶ [i]