MinT - Best Known (137−72, 137, s)-Nets in Base 2

Best Known (137−72, 137, s)-Nets in Base 2

(137−72, 137, 43)-Net over F₂ — Constructive and digital

Digital (65, 137, 43)-net over F₂, using

t-expansion [i] based on digital (59, 137, 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]

(137−72, 137, 48)-Net over F₂ — Digital

Digital (65, 137, 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]

(137−72, 137, 139)-Net over F₂ — Upper bound on s (digital)

There is no digital (65, 137, 140)-net over F₂, because

6 times m-reduction [i] would yield digital (65, 131, 140)-net over F₂, but
- extracting embedded orthogonal array [i] would yield linear OA(2¹³¹, 140, F₂, 66) (dual of [140, 9, 67]-code), but
  - residual code [i] would yield linear OA(2⁶⁵, 73, F₂, 33) (dual of [73, 8, 34]-code), but
    - â€œBJVâ€ bound on codes from Brouwerâ€™s database [i]

(137−72, 137, 141)-Net in Base 2 — Upper bound on s

There is no (65, 137, 142)-net in base 2, because

6 times m-reduction [i] would yield (65, 131, 142)-net in base 2, but
- extracting embedded orthogonal array [i] would yield OA(2¹³¹, 142, S₂, 66), but
  - the linear programming bound shows that MÂ â‰¥ 10 529697 562001 519813 410663 852368 635535 294464 / 3485 > 2¹³¹ [i]