MinT - Best Known (210−112, 210, s)-Nets in Base 2

Best Known (210−112, 210, s)-Nets in Base 2

(210−112, 210, 54)-Net over F₂ — Constructive and digital

Digital (98, 210, 54)-net over F₂, using

t-expansion [i] based on digital (95, 210, 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]

(210−112, 210, 65)-Net over F₂ — Digital

Digital (98, 210, 65)-net over F₂, using

t-expansion [i] based on digital (95, 210, 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]

(210−112, 210, 206)-Net over F₂ — Upper bound on s (digital)

There is no digital (98, 210, 207)-net over F₂, because

12 times m-reduction [i] would yield digital (98, 198, 207)-net over F₂, but
- extracting embedded orthogonal array [i] would yield linear OA(2¹⁹⁸, 207, F₂, 100) (dual of [207, 9, 101]-code), but
  - residual code [i] would yield linear OA(2⁹⁸, 106, F₂, 50) (dual of [106, 8, 51]-code), but
    - residual code [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]

(210−112, 210, 208)-Net in Base 2 — Upper bound on s

There is no (98, 210, 209)-net in base 2, because

12 times m-reduction [i] would yield (98, 198, 209)-net in base 2, but
- extracting embedded orthogonal array [i] would yield OA(2¹⁹⁸, 209, S₂, 100), but
  - the linear programming bound shows that MÂ â‰¥ 1298 405939 761264 142637 905370 611659 382837 940018 976496 610923 511808 / 3111 > 2¹⁹⁸ [i]