MinT - Best Known (55, 55+68, s)-Nets in Base 2

Best Known (55, 55+68, s)-Nets in Base 2

(55, 55+68, 42)-Net over F₂ — Constructive and digital

Digital (55, 123, 42)-net over F₂, using

t-expansion [i] based on digital (54, 123, 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]

(55, 55+68, 119)-Net over F₂ — Upper bound on s (digital)

There is no digital (55, 123, 120)-net over F₂, because

12 times m-reduction [i] would yield digital (55, 111, 120)-net over F₂, but
- extracting embedded orthogonal array [i] would yield linear OA(2¹¹¹, 120, F₂, 56) (dual of [120, 9, 57]-code), but
  - residual code [i] would yield linear OA(2⁵⁵, 63, F₂, 28) (dual of [63, 8, 29]-code), but
    - â€œBJVâ€ bound on codes from Brouwerâ€™s database [i]

(55, 55+68, 120)-Net in Base 2 — Upper bound on s

There is no (55, 123, 121)-net in base 2, because

8 times m-reduction [i] would yield (55, 115, 121)-net in base 2, but
- extracting embedded orthogonal array [i] would yield OA(2¹¹⁵, 121, S₂, 60), but
  - adding a parity check bit [i] would yield OA(2¹¹⁶, 122, S₂, 61), but
    - the (dual) Plotkin bound shows that MÂ â‰¥ 2 658455 991569 831745 807614 120560 689152 / 31 > 2¹¹⁶ [i]