MinT - Best Known (164−51, 164, s)-Nets in Base 8

Best Known (164−51, 164, s)-Nets in Base 8

(164−51, 164, 1026)-Net over F₈ — Constructive and digital

Digital (113, 164, 1026)-net over F₈, using

6 times m-reduction [i] based on digital (113, 170, 1026)-net over F₈, using
- trace code for nets [i] based on digital (28, 85, 513)-net over F₆₄, using
  - net from sequence [i] based on digital (28, 512)-sequence over F₆₄, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 28 and N(F) ≥ 513, using
      - the Hermitian function field over F₆₄ [i]

(164−51, 164, 2576)-Net over F₈ — Digital

Digital (113, 164, 2576)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(164−51, 164, 1123746)-Net in Base 8 — Upper bound on s

There is no (113, 164, 1123747)-net in base 8, because

1 times m-reduction [i] would yield (113, 163, 1123747)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 1598 340854 454100 026514 356272 525343 607193 147856 646827 116286 992341 564570 773383 078089 826641 951849 685186 725088 161657 141801 798099 023628 754193 875031 005576 > 8¹⁶³ [i]