MinT - Best Known (53−29, 53, s)-Nets in Base 2

Best Known (53−29, 53, s)-Nets in Base 2

(53−29, 53, 21)-Net over F₂ — Constructive and digital

Digital (24, 53, 21)-net over F₂, using

t-expansion [i] based on digital (21, 53, 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]

(53−29, 53, 22)-Net over F₂ — Digital

Digital (24, 53, 22)-net over F₂, using

t-expansion [i] based on digital (23, 53, 22)-net over F₂, using
- net from sequence [i] based on digital (23, 21)-sequence over F₂, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂ with g(F) = 23 and N(F) ≥ 22, using
    - Drinfeld modules of rank 1 [i]

(53−29, 53, 55)-Net over F₂ — Upper bound on s (digital)

There is no digital (24, 53, 56)-net over F₂, because

5 times m-reduction [i] would yield digital (24, 48, 56)-net over F₂, but
- extracting embedded orthogonal array [i] would yield linear OA(2⁴⁸, 56, F₂, 24) (dual of [56, 8, 25]-code), but
  - adding a parity check bit [i] would yield linear OA(2⁴⁹, 57, F₂, 25) (dual of [57, 8, 26]-code), but
    - â€œBJVâ€ bound on codes from Brouwerâ€™s database [i]

(53−29, 53, 56)-Net in Base 2 — Upper bound on s

There is no (24, 53, 57)-net in base 2, because

1 times m-reduction [i] would yield (24, 52, 57)-net in base 2, but
- extracting embedded orthogonal array [i] would yield OA(2⁵², 57, S₂, 28), but
  - adding a parity check bit [i] would yield OA(2⁵³, 58, S₂, 29), but
    - the (dual) Plotkin bound shows that MÂ â‰¥ 144115 188075 855872 / 15 > 2⁵³ [i]