MinT - Best Known (163−77, 163, s)-Nets in Base 8

Best Known (163−77, 163, s)-Nets in Base 8

(163−77, 163, 208)-Net over F₈ — Constructive and digital

Digital (86, 163, 208)-net over F₈, using

3 times m-reduction [i] based on digital (86, 166, 208)-net over F₈, using
- trace code for nets [i] based on digital (3, 83, 104)-net over F₆₄, using
  - net from sequence [i] based on digital (3, 103)-sequence over F₆₄, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 3 and N(F) ≥ 104, using
      - a function field by SÃ©mirat [i]

(163−77, 163, 225)-Net in Base 8 — Constructive

(86, 163, 225)-net in base 8, using

t-expansion [i] based on (83, 163, 225)-net in base 8, using
- 9 times m-reduction [i] based on (83, 172, 225)-net in base 8, using
  - base change [i] based on digital (40, 129, 225)-net over F₁₆, using
    - net from sequence [i] based on digital (40, 224)-sequence over F₁₆, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₆ with g(F) = 40 and N(F) ≥ 225, using
        a function field by GarcÃa and Quoos [i]

(163−77, 163, 331)-Net over F₈ — Digital

Digital (86, 163, 331)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(163−77, 163, 15171)-Net in Base 8 — Upper bound on s

There is no (86, 163, 15172)-net in base 8, because

1 times m-reduction [i] would yield (86, 162, 15172)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 199 906912 449778 247127 017108 146602 458270 208447 286308 015214 977955 164361 171760 431452 219917 183878 735609 415210 320848 220523 197595 673944 831070 311900 454800 > 8¹⁶² [i]