MinT - Best Known (53−30, 53, s)-Nets in Base 2

Best Known (53−30, 53, s)-Nets in Base 2

(53−30, 53, 21)-Net over F₂ — Constructive and digital

Digital (23, 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]

(53−30, 53, 22)-Net over F₂ — Digital

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]

(53−30, 53, 53)-Net over F₂ — Upper bound on s (digital)

There is no digital (23, 53, 54)-net over F₂, because

6 times m-reduction [i] would yield digital (23, 47, 54)-net over F₂, but
- extracting embedded orthogonal array [i] would yield linear OA(2⁴⁷, 54, F₂, 24) (dual of [54, 7, 25]-code), but
  - adding a parity check bit [i] would yield linear OA(2⁴⁸, 55, F₂, 25) (dual of [55, 7, 26]-code), but
    - â€œvT4â€ bound on codes from Brouwerâ€™s database [i]

(53−30, 53, 54)-Net in Base 2 — Upper bound on s

There is no (23, 53, 55)-net in base 2, because

the generalized Rao bound for nets shows that 2^mÂ â‰¥ 9139 268213 161268 > 2⁵³ [i]