MinT - Best Known (57, 113, s)-Nets in Base 2

Best Known (57, 113, s)-Nets in Base 2

(57, 113, 42)-Net over F₂ — Constructive and digital

Digital (57, 113, 42)-net over F₂, using

t-expansion [i] based on digital (54, 113, 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]

(57, 113, 126)-Net over F₂ — Upper bound on s (digital)

There is no digital (57, 113, 127)-net over F₂, because

extracting embedded orthogonal array [i] would yield linear OA(2¹¹³, 127, F₂, 56) (dual of [127, 14, 57]-code), but
- residual code [i] would yield linear OA(2⁵⁷, 70, F₂, 28) (dual of [70, 13, 29]-code), but
  - adding a parity check bit [i] would yield linear OA(2⁵⁸, 71, F₂, 29) (dual of [71, 13, 30]-code), but
    - â€œJaâ€ bound on codes from Brouwerâ€™s database [i]

(57, 113, 127)-Net in Base 2 — Upper bound on s

There is no (57, 113, 128)-net in base 2, because

extracting embedded orthogonal array [i] would yield OA(2¹¹³, 128, S₂, 56), but
- the linear programming bound shows that MÂ â‰¥ 5948 959895 135390 989180 988498 284682 149888 / 569415 > 2¹¹³ [i]