MinT - Best Known (253−124, 253, s)-Nets in Base 2

Best Known (253−124, 253, s)-Nets in Base 2

Digital (129, 253, 57)-net over F₂, using

t-expansion [i] based on digital (110, 253, 57)-net over F₂, using
- net from sequence [i] based on digital (110, 56)-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 8 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 (129, 253, 81)-net over F₂, using

t-expansion [i] based on digital (126, 253, 81)-net over F₂, using
- net from sequence [i] based on digital (126, 80)-sequence over F₂, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂ with g(F) = 126 and N(F) ≥ 81, using
    - Drinfeld modules of rank 1 [i]

There is no digital (129, 253, 277)-net over F₂, because

extracting embedded orthogonal array [i] would yield linear OA(2²⁵³, 277, F₂, 124) (dual of [277, 24, 125]-code), but
- residual code [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 (129, 253, 285)-net in base 2, because

16 times m-reduction [i] would yield (129, 237, 285)-net in base 2, but
- extracting embedded orthogonal array [i] would yield OA(2²³⁷, 285, S₂, 108), but
  - the linear programming bound shows that MÂ â‰¥ 56343 399996 195265 169227 114420 218575 369487 305782 122681 969429 903696 483971 902670 896111 812608 / 204485 216198 046875 > 2²³⁷ [i]