MinT - Best Known (167−81, 167, s)-Nets in Base 2

Best Known (167−81, 167, s)-Nets in Base 2

Digital (86, 167, 52)-net over F₂, using

t-expansion [i] based on digital (85, 167, 52)-net over F₂, using
- net from sequence [i] based on digital (85, 51)-sequence over F₂, using
  - Niederreiterâ€“Xing sequence constructionÂ III based on the algebraic function field F/F₂ with g(F)Â =Â 69, N(F)Â =Â 48, 1 place with degree 2, and 3 places with degree 6 [i] based on function field F/F₂ with g(F) = 69 and N(F) ≥ 48, using an explicitly constructive algebraic function field [i]

Digital (86, 167, 57)-net over F₂, using

t-expansion [i] based on digital (83, 167, 57)-net over F₂, using
- net from sequence [i] based on digital (83, 56)-sequence over F₂, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂ with g(F) = 83 and N(F) ≥ 57, using
    - an algebraic function field by Auer [i]

There is no digital (86, 167, 193)-net over F₂, because

3 times m-reduction [i] would yield digital (86, 164, 193)-net over F₂, but
- extracting embedded orthogonal array [i] would yield linear OA(2¹⁶⁴, 193, F₂, 78) (dual of [193, 29, 79]-code), but
  - 2 times code embedding in larger space [i] would yield linear OA(2¹⁶⁶, 195, F₂, 78) (dual of [195, 29, 79]-code), but
    - adding a parity check bit [i] would yield linear OA(2¹⁶⁷, 196, F₂, 79) (dual of [196, 29, 80]-code), but
      - â€œBKâ€ bound on codes from Brouwerâ€™s database [i]

There is no (86, 167, 226)-net in base 2, because

1 times m-reduction [i] would yield (86, 166, 226)-net in base 2, but
- the generalized Rao bound for nets shows that 2^mÂ â‰¥ 98 763727 875331 028119 597987 932474 101084 530032 637848 > 2¹⁶⁶ [i]