MinT - Best Known (57−32, 57, s)-Nets in Base 2

Best Known (57−32, 57, s)-Nets in Base 2

(57−32, 57, 21)-Net over F₂ — Constructive and digital

Digital (25, 57, 21)-net over F₂, using

t-expansion [i] based on digital (21, 57, 21)-net over F₂, using
- net from sequence [i] based on digital (21, 20)-sequence over F₂, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂ with g(F) = 21 and N(F) ≥ 21, using
    - an explicitly constructive algebraic function field [i]

(57−32, 57, 24)-Net over F₂ — Digital

Digital (25, 57, 24)-net over F₂, using

net from sequence [i] based on digital (25, 23)-sequence over F₂, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂ with g(F) = 25 and N(F) ≥ 24, using
  - Drinfeld modules of rank 1 [i]

(57−32, 57, 58)-Net over F₂ — Upper bound on s (digital)

There is no digital (25, 57, 59)-net over F₂, because

6 times m-reduction [i] would yield digital (25, 51, 59)-net over F₂, but
- extracting embedded orthogonal array [i] would yield linear OA(2⁵¹, 59, F₂, 26) (dual of [59, 8, 27]-code), but
  - adding a parity check bit [i] would yield linear OA(2⁵², 60, F₂, 27) (dual of [60, 8, 28]-code), but
    - â€œBJVâ€ bound on codes from Brouwerâ€™s database [i]

(57−32, 57, 59)-Net in Base 2 — Upper bound on s

There is no (25, 57, 60)-net in base 2, because

the generalized Rao bound for nets shows that 2^mÂ â‰¥ 157943 143978 216945 > 2⁵⁷ [i]