MinT - Best Known (23, 47, s)-Nets in Base 2

Best Known (23, 47, s)-Nets in Base 2

(23, 47, 21)-Net over F₂ — Constructive and digital

Digital (23, 47, 21)-net over F₂, using

t-expansion [i] based on digital (21, 47, 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]

(23, 47, 22)-Net over F₂ — Digital

Digital (23, 47, 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]

(23, 47, 53)-Net over F₂ — Upper bound on s (digital)

There is no digital (23, 47, 54)-net over F₂, because

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]

(23, 47, 58)-Net in Base 2 — Upper bound on s

There is no (23, 47, 59)-net in base 2, because

extracting embedded orthogonal array [i] would yield OA(2⁴⁷, 59, S₂, 24), but
- the linear programming bound shows that MÂ â‰¥ 151433 537470 332928 / 1071 > 2⁴⁷ [i]