MinT - Best Known (66, 66+69, s)-Nets in Base 2

Best Known (66, 66+69, s)-Nets in Base 2

(66, 66+69, 43)-Net over F₂ — Constructive and digital

Digital (66, 135, 43)-net over F₂, using

t-expansion [i] based on digital (59, 135, 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+69, 48)-Net over F₂ — Digital

Digital (66, 135, 48)-net over F₂, using

t-expansion [i] based on digital (65, 135, 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+69, 141)-Net over F₂ — Upper bound on s (digital)

There is no digital (66, 135, 142)-net over F₂, because

1 times m-reduction [i] would yield digital (66, 134, 142)-net over F₂, but
- extracting embedded orthogonal array [i] would yield linear OA(2¹³⁴, 142, F₂, 68) (dual of [142, 8, 69]-code), but
  - residual code [i] would yield linear OA(2⁶⁶, 73, F₂, 34) (dual of [73, 7, 35]-code), but
    - â€œHelâ€ bound on codes from Brouwerâ€™s database [i]

(66, 66+69, 143)-Net in Base 2 — Upper bound on s

There is no (66, 135, 144)-net in base 2, because

3 times m-reduction [i] would yield (66, 132, 144)-net in base 2, but
- extracting embedded orthogonal array [i] would yield OA(2¹³², 144, S₂, 66), but
  - the linear programming bound shows that MÂ â‰¥ 4 616951 154383 293072 271066 673634 231093 035008 / 697 > 2¹³² [i]