MinT - Best Known (66, 66+59, s)-Nets in Base 2

Best Known (66, 66+59, s)-Nets in Base 2

(66, 66+59, 43)-Net over F₂ — Constructive and digital

Digital (66, 125, 43)-net over F₂, using

t-expansion [i] based on digital (59, 125, 43)-net over F₂, using
- net from sequence [i] based on digital (59, 42)-sequence over F₂, using
  - Niederreiterâ€“Xing sequence constructionÂ III based on the algebraic function field F/F₂ with g(F)Â =Â 54, N(F)Â =Â 42, and 1 place with degree 6 [i] based on function field F/F₂ with g(F) = 54 and N(F) ≥ 42, using an explicitly constructive algebraic function field [i]

(66, 66+59, 48)-Net over F₂ — Digital

Digital (66, 125, 48)-net over F₂, using

t-expansion [i] based on digital (65, 125, 48)-net over F₂, using
- net from sequence [i] based on digital (65, 47)-sequence over F₂, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂ with g(F) = 65 and N(F) ≥ 48, using
    - Drinfeld modules of rank 1 [i]

(66, 66+59, 165)-Net over F₂ — Upper bound on s (digital)

There is no digital (66, 125, 166)-net over F₂, because

1 times m-reduction [i] would yield digital (66, 124, 166)-net over F₂, but
- extracting embedded orthogonal array [i] would yield linear OA(2¹²⁴, 166, F₂, 58) (dual of [166, 42, 59]-code), but
  - constructionÂ Y1 [i] would yield
    1. OA(2¹²³, 150, S₂, 58), but
      - the linear programming bound shows that MÂ â‰¥ 36000 467012 365704 432940 148948 904738 978722 217984 / 3201 323125 > 2¹²³ [i]
    2. OA(2⁴², 166, S₂, 16), but
      - discarding factors would yield OA(2⁴², 146, S₂, 16), but
        the Rao or (dual) Hamming bound shows that MÂ â‰¥ 4 467697 956730 > 2⁴² [i]

(66, 66+59, 186)-Net in Base 2 — Upper bound on s

There is no (66, 125, 187)-net in base 2, because

1 times m-reduction [i] would yield (66, 124, 187)-net in base 2, but
- the generalized Rao bound for nets shows that 2^mÂ â‰¥ 22 765626 897036 575690 075424 768676 037072 > 2¹²⁴ [i]