MinT - Best Known (197−94, 197, s)-Nets in Base 2

Best Known (197−94, 197, s)-Nets in Base 2

(197−94, 197, 55)-Net over F₂ — Constructive and digital

Digital (103, 197, 55)-net over F₂, using

t-expansion [i] based on digital (100, 197, 55)-net over F₂, using
- net from sequence [i] based on digital (100, 54)-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 6 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]

(197−94, 197, 65)-Net over F₂ — Digital

Digital (103, 197, 65)-net over F₂, using

t-expansion [i] based on digital (95, 197, 65)-net over F₂, using
- net from sequence [i] based on digital (95, 64)-sequence over F₂, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂ with g(F) = 95 and N(F) ≥ 65, using
    - Drinfeld modules of rank 1 [i]

(197−94, 197, 231)-Net over F₂ — Upper bound on s (digital)

There is no digital (103, 197, 232)-net over F₂, because

2 times m-reduction [i] would yield digital (103, 195, 232)-net over F₂, but
- extracting embedded orthogonal array [i] would yield linear OA(2¹⁹⁵, 232, F₂, 92) (dual of [232, 37, 93]-code), but
  - constructionÂ Y1 [i] would yield
    1. linear OA(2¹⁹⁴, 220, F₂, 92) (dual of [220, 26, 93]-code), but
      - adding a parity check bit [i] would yield linear OA(2¹⁹⁵, 221, F₂, 93) (dual of [221, 26, 94]-code), but
        â€œBKâ€ bound on codes from Brouwerâ€™s database [i]
    2. OA(2³⁷, 232, S₂, 12), but
      - discarding factors would yield OA(2³⁷, 217, S₂, 12), but
        the Rao or (dual) Hamming bound shows that MÂ â‰¥ 139173 045698 > 2³⁷ [i]

(197−94, 197, 271)-Net in Base 2 — Upper bound on s

There is no (103, 197, 272)-net in base 2, because

the generalized Rao bound for nets shows that 2^mÂ â‰¥ 211658 797210 701179 192075 412282 244676 735345 976550 576201 896936 > 2¹⁹⁷ [i]