MinT - Best Known (155−55, 155, s)-Nets in Base 8

Best Known (155−55, 155, s)-Nets in Base 8

(155−55, 155, 379)-Net over F₈ — Constructive and digital

Digital (100, 155, 379)-net over F₈, using

(u,Â u+v)-construction [i] based on
1. digital (4, 31, 25)-net over F₈, using
  - net from sequence [i] based on digital (4, 24)-sequence over F₈, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈ with g(F) = 4 and N(F) ≥ 25, using
      - a function field by SÃ©mirat [i]
2. digital (69, 124, 354)-net over F₈, using
  - trace code for nets [i] based on digital (7, 62, 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]

(155−55, 155, 576)-Net in Base 8 — Constructive

(100, 155, 576)-net in base 8, using

3 times m-reduction [i] based on (100, 158, 576)-net in base 8, using
- trace code for nets [i] based on (21, 79, 288)-net in base 64, using
  - 5 times m-reduction [i] based on (21, 84, 288)-net in base 64, using
    - base change [i] based on digital (9, 72, 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]

(155−55, 155, 1198)-Net over F₈ — Digital

Digital (100, 155, 1198)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(155−55, 155, 220919)-Net in Base 8 — Upper bound on s

There is no (100, 155, 220920)-net in base 8, because

1 times m-reduction [i] would yield (100, 154, 220920)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 11 909812 037852 299225 718463 620563 028803 451479 989909 692806 628892 021202 319991 759206 451110 930334 356231 200118 733635 119127 698631 893650 236779 648620 > 8¹⁵⁴ [i]