MinT - Best Known (67, 67+63, s)-Nets in Base 2

Best Known (67, 67+63, s)-Nets in Base 2

Digital (67, 130, 43)-net over F₂, using

t-expansion [i] based on digital (59, 130, 43)-net over F₂, using
- net from sequence [i] based on digital (59, 42)-sequence over F₂, using
  - Niederreiterâ€“Xing sequence constructionÂ III based on the algebraic function field F/F₂ with g(F)Â =Â 54, N(F)Â =Â 42, and 1 place with degree 6 [i] based on function field F/F₂ with g(F) = 54 and N(F) ≥ 42, using an explicitly constructive algebraic function field [i]

Digital (67, 130, 48)-net over F₂, using

t-expansion [i] based on digital (65, 130, 48)-net over F₂, using
- net from sequence [i] based on digital (65, 47)-sequence over F₂, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂ with g(F) = 65 and N(F) ≥ 48, using
    - Drinfeld modules of rank 1 [i]

There is no digital (67, 130, 152)-net over F₂, because

1 times m-reduction [i] would yield digital (67, 129, 152)-net over F₂, but
- extracting embedded orthogonal array [i] would yield linear OA(2¹²⁹, 152, F₂, 62) (dual of [152, 23, 63]-code), but
  - 1 times code embedding in larger space [i] would yield linear OA(2¹³⁰, 153, F₂, 62) (dual of [153, 23, 63]-code), but
    - adding a parity check bit [i] would yield linear OA(2¹³¹, 154, F₂, 63) (dual of [154, 23, 64]-code), but
      - â€œBKâ€ bound on codes from Brouwerâ€™s database [i]

There is no (67, 130, 181)-net in base 2, because

1 times m-reduction [i] would yield (67, 129, 181)-net in base 2, but
- the generalized Rao bound for nets shows that 2^mÂ â‰¥ 773 901620 193970 258084 211259 472490 874464 > 2¹²⁹ [i]