MinT - Best Known (81, 81+78, s)-Nets in Base 2

Best Known (81, 81+78, s)-Nets in Base 2

(81, 81+78, 51)-Net over F₂ — Constructive and digital

Digital (81, 159, 51)-net over F₂, using

t-expansion [i] based on digital (80, 159, 51)-net over F₂, using
- net from sequence [i] based on digital (80, 50)-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 2 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]

(81, 81+78, 56)-Net over F₂ — Digital

Digital (81, 159, 56)-net over F₂, using

t-expansion [i] based on digital (80, 159, 56)-net over F₂, using
- net from sequence [i] based on digital (80, 55)-sequence over F₂, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂ with g(F) = 80 and N(F) ≥ 56, using
    - Drinfeld modules of rank 1 [i]

(81, 81+78, 177)-Net over F₂ — Upper bound on s (digital)

There is no digital (81, 159, 178)-net over F₂, because

4 times m-reduction [i] would yield digital (81, 155, 178)-net over F₂, but
- extracting embedded orthogonal array [i] would yield linear OA(2¹⁵⁵, 178, F₂, 74) (dual of [178, 23, 75]-code), but
  - adding a parity check bit [i] would yield linear OA(2¹⁵⁶, 179, F₂, 75) (dual of [179, 23, 76]-code), but
    - â€œBKâ€ bound on codes from Brouwerâ€™s database [i]

(81, 81+78, 207)-Net in Base 2 — Upper bound on s

There is no (81, 159, 208)-net in base 2, because

the generalized Rao bound for nets shows that 2^mÂ â‰¥ 835992 238376 411240 598676 195541 388528 788944 376610 > 2¹⁵⁹ [i]