MinT - Best Known (139−55, 139, s)-Nets in Base 8

Best Known (139−55, 139, s)-Nets in Base 8

(139−55, 139, 354)-Net over F₈ — Constructive and digital

Digital (84, 139, 354)-net over F₈, using

15 times m-reduction [i] based on digital (84, 154, 354)-net over F₈, using
- trace code for nets [i] based on digital (7, 77, 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]

(139−55, 139, 384)-Net in Base 8 — Constructive

(84, 139, 384)-net in base 8, using

t-expansion [i] based on (83, 139, 384)-net in base 8, using
- 1 times m-reduction [i] based on (83, 140, 384)-net in base 8, using
  - trace code for nets [i] based on (13, 70, 192)-net in base 64, using
    - base change [i] based on digital (3, 60, 192)-net over F₁₂₈, using
      - net from sequence [i] based on digital (3, 191)-sequence over F₁₂₈, using
        Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₂₈ with g(F) = 3 and N(F) ≥ 192, using
        a function field by SÃ©mirat [i]

(139−55, 139, 633)-Net over F₈ — Digital

Digital (84, 139, 633)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(139−55, 139, 64415)-Net in Base 8 — Upper bound on s

There is no (84, 139, 64416)-net in base 8, because

1 times m-reduction [i] would yield (84, 138, 64416)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 42321 744699 120935 395365 295174 300687 386940 834231 299412 215618 593058 938602 039839 964766 602333 770486 790619 908476 865213 063684 632575 > 8¹³⁸ [i]