MinT - Best Known (170−81, 170, s)-Nets in Base 8

Best Known (170−81, 170, s)-Nets in Base 8

(170−81, 170, 208)-Net over F₈ — Constructive and digital

Digital (89, 170, 208)-net over F₈, using

2 times m-reduction [i] based on digital (89, 172, 208)-net over F₈, using
- trace code for nets [i] based on digital (3, 86, 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]

(170−81, 170, 225)-Net in Base 8 — Constructive

(89, 170, 225)-net in base 8, using

t-expansion [i] based on (83, 170, 225)-net in base 8, using
- 2 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]

(170−81, 170, 331)-Net over F₈ — Digital

Digital (89, 170, 331)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(170−81, 170, 14706)-Net in Base 8 — Upper bound on s

There is no (89, 170, 14707)-net in base 8, because

1 times m-reduction [i] would yield (89, 169, 14707)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 419 741689 672582 327158 559307 055051 738219 986202 060284 649051 042494 030471 785961 165940 077433 638618 654227 667495 258172 670380 158360 321608 703357 611103 908476 503934 > 8¹⁶⁹ [i]