MinT - Best Known (188−95, 188, s)-Nets in Base 2

Best Known (188−95, 188, s)-Nets in Base 2

(188−95, 188, 53)-Net over F₂ — Constructive and digital

Digital (93, 188, 53)-net over F₂, using

t-expansion [i] based on digital (90, 188, 53)-net over F₂, using
- net from sequence [i] based on digital (90, 52)-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 4 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]

(188−95, 188, 60)-Net over F₂ — Digital

Digital (93, 188, 60)-net over F₂, using

t-expansion [i] based on digital (92, 188, 60)-net over F₂, using
- net from sequence [i] based on digital (92, 59)-sequence over F₂, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂ with g(F) = 92 and N(F) ≥ 60, using
    - Drinfeld modules of rank 1 [i]

(188−95, 188, 198)-Net over F₂ — Upper bound on s (digital)

There is no digital (93, 188, 199)-net over F₂, because

1 times m-reduction [i] would yield digital (93, 187, 199)-net over F₂, but
- extracting embedded orthogonal array [i] would yield linear OA(2¹⁸⁷, 199, F₂, 94) (dual of [199, 12, 95]-code), but
  - residual code [i] would yield linear OA(2⁹³, 104, F₂, 47) (dual of [104, 11, 48]-code), but
    - â€œJaâ€ bound on codes from Brouwerâ€™s database [i]

(188−95, 188, 226)-Net in Base 2 — Upper bound on s

There is no (93, 188, 227)-net in base 2, because

1 times m-reduction [i] would yield (93, 187, 227)-net in base 2, but
- the generalized Rao bound for nets shows that 2^mÂ â‰¥ 214 622485 842892 703917 487597 559316 923108 587610 787300 089568 > 2¹⁸⁷ [i]