MinT - Best Known (53−26, 53, s)-Nets in Base 2

Best Known (53−26, 53, s)-Nets in Base 2

(53−26, 53, 21)-Net over F₂ — Constructive and digital

Digital (27, 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]

(53−26, 53, 24)-Net over F₂ — Digital

Digital (27, 53, 24)-net over F₂, using

t-expansion [i] based on digital (25, 53, 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]

(53−26, 53, 67)-Net over F₂ — Upper bound on s (digital)

There is no digital (27, 53, 68)-net over F₂, because

extracting embedded orthogonal array [i] would yield linear OA(2⁵³, 68, F₂, 26) (dual of [68, 15, 27]-code), but
- â€œJaâ€ bound on codes from Brouwerâ€™s database [i]

(53−26, 53, 70)-Net in Base 2 — Upper bound on s

There is no (27, 53, 71)-net in base 2, because

extracting embedded orthogonal array [i] would yield OA(2⁵³, 71, S₂, 26), but
- the linear programming bound shows that MÂ â‰¥ 848 550227 390639 374336 / 85085 > 2⁵³ [i]