MinT - Best Known (207−104, 207, s)-Nets in Base 2

Best Known (207−104, 207, s)-Nets in Base 2

Digital (103, 207, 55)-net over F₂, using

t-expansion [i] based on digital (100, 207, 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]

Digital (103, 207, 65)-net over F₂, using

t-expansion [i] based on digital (95, 207, 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]

There is no digital (103, 207, 217)-net over F₂, because

extracting embedded orthogonal array [i] would yield linear OA(2²⁰⁷, 217, F₂, 104) (dual of [217, 10, 105]-code), but
- residual code [i] would yield linear OA(2¹⁰³, 112, F₂, 52) (dual of [112, 9, 53]-code), but
  - residual code [i] would yield linear OA(2⁵¹, 59, F₂, 26) (dual of [59, 8, 27]-code), but
    - adding a parity check bit [i] would yield linear OA(2⁵², 60, F₂, 27) (dual of [60, 8, 28]-code), but
      - â€œBJVâ€ bound on codes from Brouwerâ€™s database [i]

There is no (103, 207, 218)-net in base 2, because

2 times m-reduction [i] would yield (103, 205, 218)-net in base 2, but
- extracting embedded orthogonal array [i] would yield OA(2²⁰⁵, 218, S₂, 102), but
  - the linear programming bound shows that MÂ â‰¥ 272331 004236 659599 976647 399713 241508 574642 785762 357467 384194 793472 / 3835 > 2²⁰⁵ [i]