MinT - Best Known (204−109, 204, s)-Nets in Base 2

Best Known (204−109, 204, s)-Nets in Base 2

(204−109, 204, 54)-Net over F₂ — Constructive and digital

Digital (95, 204, 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]

(204−109, 204, 65)-Net over F₂ — Digital

Digital (95, 204, 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]

(204−109, 204, 199)-Net over F₂ — Upper bound on s (digital)

There is no digital (95, 204, 200)-net over F₂, because

13 times m-reduction [i] would yield digital (95, 191, 200)-net over F₂, but
- extracting embedded orthogonal array [i] would yield linear OA(2¹⁹¹, 200, F₂, 96) (dual of [200, 9, 97]-code), but
  - residual code [i] would yield linear OA(2⁹⁵, 103, F₂, 48) (dual of [103, 8, 49]-code), but
    - residual code [i] would yield linear OA(2⁴⁷, 54, F₂, 24) (dual of [54, 7, 25]-code), but
      - adding a parity check bit [i] would yield linear OA(2⁴⁸, 55, F₂, 25) (dual of [55, 7, 26]-code), but
        â€œvT4â€ bound on codes from Brouwerâ€™s database [i]

(204−109, 204, 202)-Net in Base 2 — Upper bound on s

There is no (95, 204, 203)-net in base 2, because

7 times m-reduction [i] would yield (95, 197, 203)-net in base 2, but
- extracting embedded orthogonal array [i] would yield OA(2¹⁹⁷, 203, S₂, 102), but
  - adding a parity check bit [i] would yield OA(2¹⁹⁸, 204, S₂, 103), but
    - the (dual) Plotkin bound shows that MÂ â‰¥ 6 427752 177035 961102 167848 369364 650410 088811 975131 171341 205504 / 13 > 2¹⁹⁸ [i]