MinT - Best Known (252−131, 252, s)-Nets in Base 2

Best Known (252−131, 252, s)-Nets in Base 2

(252−131, 252, 57)-Net over F₂ — Constructive and digital

Digital (121, 252, 57)-net over F₂, using

t-expansion [i] based on digital (110, 252, 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]

(252−131, 252, 80)-Net over F₂ — Digital

Digital (121, 252, 80)-net over F₂, using

net from sequence [i] based on digital (121, 79)-sequence over F₂, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂ with g(F) = 121 and N(F) ≥ 80, using
  - Drinfeld modules of rank 1 [i]

(252−131, 252, 254)-Net over F₂ — Upper bound on s (digital)

There is no digital (121, 252, 255)-net over F₂, because

7 times m-reduction [i] would yield digital (121, 245, 255)-net over F₂, but
- extracting embedded orthogonal array [i] would yield linear OA(2²⁴⁵, 255, F₂, 124) (dual of [255, 10, 125]-code), but
  - residual code [i] would yield linear OA(2¹²¹, 130, F₂, 62) (dual of [130, 9, 63]-code), but
    - residual code [i] would yield linear OA(2⁵⁹, 67, F₂, 31) (dual of [67, 8, 32]-code), but
      - â€œDHMâ€ bound on codes from Brouwerâ€™s database [i]

(252−131, 252, 255)-Net in Base 2 — Upper bound on s

There is no (121, 252, 256)-net in base 2, because

3 times m-reduction [i] would yield (121, 249, 256)-net in base 2, but
- extracting embedded orthogonal array [i] would yield OA(2²⁴⁹, 256, S₂, 128), but
  - adding a parity check bit [i] would yield OA(2²⁵⁰, 257, S₂, 129), but
    - the (dual) Plotkin bound shows that MÂ â‰¥ 173688 133855 974293 135356 477513 031861 779904 976998 460846 059186 376011 869694 459904 / 65 > 2²⁵⁰ [i]