MinT - Best Known (113−58, 113, s)-Nets in Base 2

Best Known (113−58, 113, s)-Nets in Base 2

(113−58, 113, 42)-Net over F₂ — Constructive and digital

Digital (55, 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]

(113−58, 113, 119)-Net over F₂ — Upper bound on s (digital)

There is no digital (55, 113, 120)-net over F₂, because

2 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]

(113−58, 113, 121)-Net in Base 2 — Upper bound on s

There is no (55, 113, 122)-net in base 2, because

2 times m-reduction [i] would yield (55, 111, 122)-net in base 2, but
- extracting embedded orthogonal array [i] would yield OA(2¹¹¹, 122, S₂, 56), but
  - the linear programming bound shows that MÂ â‰¥ 53 252196 581133 192158 208770 352481 304576 / 17255 > 2¹¹¹ [i]