MinT - Best Known (112−56, 112, s)-Nets in Base 2

Best Known (112−56, 112, s)-Nets in Base 2

(112−56, 112, 42)-Net over F₂ — Constructive and digital

Digital (56, 112, 42)-net over F₂, using

t-expansion [i] based on digital (54, 112, 42)-net over F₂, using
- net from sequence [i] based on digital (54, 41)-sequence over F₂, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂ with g(F) = 54 and N(F) ≥ 42, using
    - an explicitly constructive algebraic function field [i]

(112−56, 112, 122)-Net over F₂ — Upper bound on s (digital)

There is no digital (56, 112, 123)-net over F₂, because

extracting embedded orthogonal array [i] would yield linear OA(2¹¹², 123, F₂, 56) (dual of [123, 11, 57]-code), but
- residual code [i] would yield linear OA(2⁵⁶, 66, F₂, 28) (dual of [66, 10, 29]-code), but
  - adding a parity check bit [i] would yield linear OA(2⁵⁷, 67, F₂, 29) (dual of [67, 10, 30]-code), but
    - â€œJaâ€ bound on codes from Brouwerâ€™s database [i]

(112−56, 112, 123)-Net in Base 2 — Upper bound on s

There is no (56, 112, 124)-net in base 2, because

extracting embedded orthogonal array [i] would yield OA(2¹¹², 124, S₂, 56), but
- the linear programming bound shows that MÂ â‰¥ 21 267647 932558 653966 460912 964485 513216 / 3451 > 2¹¹² [i]