MinT - Best Known (209−110, 209, s)-Nets in Base 2

Best Known (209−110, 209, s)-Nets in Base 2

(209−110, 209, 54)-Net over F₂ — Constructive and digital

Digital (99, 209, 54)-net over F₂, using

t-expansion [i] based on digital (95, 209, 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]

(209−110, 209, 65)-Net over F₂ — Digital

Digital (99, 209, 65)-net over F₂, using

t-expansion [i] based on digital (95, 209, 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]

(209−110, 209, 208)-Net over F₂ — Upper bound on s (digital)

There is no digital (99, 209, 209)-net over F₂, because

10 times m-reduction [i] would yield digital (99, 199, 209)-net over F₂, but
- extracting embedded orthogonal array [i] would yield linear OA(2¹⁹⁹, 209, F₂, 100) (dual of [209, 10, 101]-code), but
  - residual code [i] would yield linear OA(2⁹⁹, 108, F₂, 50) (dual of [108, 9, 51]-code), but
    - residual code [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]

(209−110, 209, 210)-Net in Base 2 — Upper bound on s

There is no (99, 209, 211)-net in base 2, because

10 times m-reduction [i] would yield (99, 199, 211)-net in base 2, but
- extracting embedded orthogonal array [i] would yield OA(2¹⁹⁹, 211, S₂, 100), but
  - the linear programming bound shows that MÂ â‰¥ 50804 953207 292236 551534 673511 458196 841341 969851 436778 280888 303616 / 45135 > 2¹⁹⁹ [i]