MinT - Best Known (162−79, 162, s)-Nets in Base 8

Best Known (162−79, 162, s)-Nets in Base 8

(162−79, 162, 160)-Net over F₈ — Constructive and digital

Digital (83, 162, 160)-net over F₈, using

2 times m-reduction [i] based on digital (83, 164, 160)-net over F₈, using
- trace code for nets [i] based on digital (1, 82, 80)-net over F₆₄, using
  - net from sequence [i] based on digital (1, 79)-sequence over F₆₄, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 1 and N(F) ≥ 80, using
      - a function field by SÃ©mirat [i]

(162−79, 162, 225)-Net in Base 8 — Constructive

(83, 162, 225)-net in base 8, using

10 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]

(162−79, 162, 289)-Net over F₈ — Digital

Digital (83, 162, 289)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(162−79, 162, 11737)-Net in Base 8 — Upper bound on s

There is no (83, 162, 11738)-net in base 8, because

1 times m-reduction [i] would yield (83, 161, 11738)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 25 049820 323298 250815 254488 311410 471670 049942 384345 393575 190320 368187 777286 453664 520302 107902 269144 492823 006481 149455 637976 754496 896170 239788 034488 > 8¹⁶¹ [i]