MinT - Best Known (93, 93+91, s)-Nets in Base 2

Best Known (93, 93+91, s)-Nets in Base 2

(93, 93+91, 53)-Net over F₂ — Constructive and digital

Digital (93, 184, 53)-net over F₂, using

t-expansion [i] based on digital (90, 184, 53)-net over F₂, using
- net from sequence [i] based on digital (90, 52)-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 4 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]

(93, 93+91, 60)-Net over F₂ — Digital

Digital (93, 184, 60)-net over F₂, using

t-expansion [i] based on digital (92, 184, 60)-net over F₂, using
- net from sequence [i] based on digital (92, 59)-sequence over F₂, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂ with g(F) = 92 and N(F) ≥ 60, using
    - Drinfeld modules of rank 1 [i]

(93, 93+91, 205)-Net over F₂ — Upper bound on s (digital)

There is no digital (93, 184, 206)-net over F₂, because

1 times m-reduction [i] would yield digital (93, 183, 206)-net over F₂, but
- extracting embedded orthogonal array [i] would yield linear OA(2¹⁸³, 206, F₂, 90) (dual of [206, 23, 91]-code), but
  - residual code [i] would yield OA(2⁹³, 115, S₂, 45), but
    - 1 times truncation [i] would yield OA(2⁹², 114, S₂, 44), but
      - the linear programming bound shows that MÂ â‰¥ 26 699890 767307 081769 024311 263232 / 4557 > 2⁹² [i]

(93, 93+91, 234)-Net in Base 2 — Upper bound on s

There is no (93, 184, 235)-net in base 2, because

1 times m-reduction [i] would yield (93, 183, 235)-net in base 2, but
- the generalized Rao bound for nets shows that 2^mÂ â‰¥ 13 128337 809437 634279 340777 247736 101647 209628 051327 955728 > 2¹⁸³ [i]