MinT - Best Known (189−102, 189, s)-Nets in Base 2

Best Known (189−102, 189, s)-Nets in Base 2

(189−102, 189, 52)-Net over F₂ — Constructive and digital

Digital (87, 189, 52)-net over F₂, using

t-expansion [i] based on digital (85, 189, 52)-net over F₂, using
- net from sequence [i] based on digital (85, 51)-sequence over F₂, using
  - Niederreiterâ€“Xing sequence constructionÂ III based on the algebraic function field F/F₂ with g(F)Â =Â 69, N(F)Â =Â 48, 1 place with degree 2, and 3 places with degree 6 [i] based on function field F/F₂ with g(F) = 69 and N(F) ≥ 48, using an explicitly constructive algebraic function field [i]

(189−102, 189, 57)-Net over F₂ — Digital

Digital (87, 189, 57)-net over F₂, using

t-expansion [i] based on digital (83, 189, 57)-net over F₂, using
- net from sequence [i] based on digital (83, 56)-sequence over F₂, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂ with g(F) = 83 and N(F) ≥ 57, using
    - an algebraic function field by Auer [i]

(189−102, 189, 184)-Net over F₂ — Upper bound on s (digital)

There is no digital (87, 189, 185)-net over F₂, because

14 times m-reduction [i] would yield digital (87, 175, 185)-net over F₂, but
- extracting embedded orthogonal array [i] would yield linear OA(2¹⁷⁵, 185, F₂, 88) (dual of [185, 10, 89]-code), but
  - residual code [i] would yield linear OA(2⁸⁷, 96, F₂, 44) (dual of [96, 9, 45]-code), but
    - â€œJaâ€ bound on codes from Brouwerâ€™s database [i]

(189−102, 189, 186)-Net in Base 2 — Upper bound on s

There is no (87, 189, 187)-net in base 2, because

8 times m-reduction [i] would yield (87, 181, 187)-net in base 2, but
- extracting embedded orthogonal array [i] would yield OA(2¹⁸¹, 187, S₂, 94), but
  - the (dual) Plotkin bound shows that MÂ â‰¥ 294 239143 846250 660804 802629 212859 363254 797911 459251 617792 / 95 > 2¹⁸¹ [i]