MinT - Best Known (169−81, 169, s)-Nets in Base 2

Best Known (169−81, 169, s)-Nets in Base 2

(169−81, 169, 52)-Net over F₂ — Constructive and digital

Digital (88, 169, 52)-net over F₂, using

t-expansion [i] based on digital (85, 169, 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]

(169−81, 169, 57)-Net over F₂ — Digital

Digital (88, 169, 57)-net over F₂, using

t-expansion [i] based on digital (83, 169, 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]

(169−81, 169, 194)-Net over F₂ — Upper bound on s (digital)

There is no digital (88, 169, 195)-net over F₂, because

3 times m-reduction [i] would yield digital (88, 166, 195)-net over F₂, but
- extracting embedded orthogonal array [i] would yield linear OA(2¹⁶⁶, 195, F₂, 78) (dual of [195, 29, 79]-code), but
  - adding a parity check bit [i] would yield linear OA(2¹⁶⁷, 196, F₂, 79) (dual of [196, 29, 80]-code), but
    - â€œBKâ€ bound on codes from Brouwerâ€™s database [i]

(169−81, 169, 235)-Net in Base 2 — Upper bound on s

There is no (88, 169, 236)-net in base 2, because

1 times m-reduction [i] would yield (88, 168, 236)-net in base 2, but
- the generalized Rao bound for nets shows that 2^mÂ â‰¥ 410 480005 461478 652393 799102 954935 673675 352511 178063 > 2¹⁶⁸ [i]