MinT - Best Known (196−100, 196, s)-Nets in Base 2

Best Known (196−100, 196, s)-Nets in Base 2

(196−100, 196, 54)-Net over F₂ — Constructive and digital

Digital (96, 196, 54)-net over F₂, using

t-expansion [i] based on digital (95, 196, 54)-net over F₂, using
- net from sequence [i] based on digital (95, 53)-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 5 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]

(196−100, 196, 65)-Net over F₂ — Digital

Digital (96, 196, 65)-net over F₂, using

t-expansion [i] based on digital (95, 196, 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]

(196−100, 196, 201)-Net over F₂ — Upper bound on s (digital)

There is no digital (96, 196, 202)-net over F₂, because

4 times m-reduction [i] would yield digital (96, 192, 202)-net over F₂, but
- extracting embedded orthogonal array [i] would yield linear OA(2¹⁹², 202, F₂, 96) (dual of [202, 10, 97]-code), but
  - residual code [i] would yield linear OA(2⁹⁶, 105, F₂, 48) (dual of [105, 9, 49]-code), but
    - residual code [i] would yield linear OA(2⁴⁸, 56, F₂, 24) (dual of [56, 8, 25]-code), but
      - adding a parity check bit [i] would yield linear OA(2⁴⁹, 57, F₂, 25) (dual of [57, 8, 26]-code), but
        â€œBJVâ€ bound on codes from Brouwerâ€™s database [i]

(196−100, 196, 206)-Net in Base 2 — Upper bound on s

There is no (96, 196, 207)-net in base 2, because

extracting embedded orthogonal array [i] would yield OA(2¹⁹⁶, 207, S₂, 100), but
- the linear programming bound shows that MÂ â‰¥ 5142 201741 628768 881734 278695 491720 328071 049580 104937 072964 403200 / 33201 > 2¹⁹⁶ [i]