MinT - Best Known (111, 164, s)-Nets in Base 8

Best Known (111, 164, s)-Nets in Base 8

(111, 164, 1026)-Net over F₈ — Constructive and digital

Digital (111, 164, 1026)-net over F₈, using

2 times m-reduction [i] based on digital (111, 166, 1026)-net over F₈, using
- trace code for nets [i] based on digital (28, 83, 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]

(111, 164, 2063)-Net over F₈ — Digital

Digital (111, 164, 2063)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(111, 164, 691614)-Net in Base 8 — Upper bound on s

There is no (111, 164, 691615)-net in base 8, because

1 times m-reduction [i] would yield (111, 163, 691615)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 1598 340231 294150 078928 972709 977375 144706 951011 418495 589567 196747 292140 154690 741334 019212 986273 720630 917410 085156 950164 041845 837645 320723 474323 835381 > 8¹⁶³ [i]