MinT - Best Known (144−70, 144, s)-Nets in Base 2

Best Known (144−70, 144, s)-Nets in Base 2

(144−70, 144, 49)-Net over F₂ — Constructive and digital

Digital (74, 144, 49)-net over F₂, using

t-expansion [i] based on digital (70, 144, 49)-net over F₂, using
- net from sequence [i] based on digital (70, 48)-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, and 1 place with degree 2 [i] based on function field F/F₂ with g(F) = 69 and N(F) ≥ 48, using an explicitly constructive algebraic function field [i]

(144−70, 144, 171)-Net over F₂ — Upper bound on s (digital)

There is no digital (74, 144, 172)-net over F₂, because

extracting embedded orthogonal array [i] would yield linear OA(2¹⁴⁴, 172, F₂, 70) (dual of [172, 28, 71]-code), but
- 4 times code embedding in larger space [i] would yield linear OA(2¹⁴⁸, 176, F₂, 70) (dual of [176, 28, 71]-code), but
  - adding a parity check bit [i] would yield linear OA(2¹⁴⁹, 177, F₂, 71) (dual of [177, 28, 72]-code), but
    - â€œBKâ€ bound on codes from Brouwerâ€™s database [i]

(144−70, 144, 193)-Net in Base 2 — Upper bound on s

There is no (74, 144, 194)-net in base 2, because

the generalized Rao bound for nets shows that 2^mÂ â‰¥ 24 131703 366748 957754 995562 713595 329328 933256 > 2¹⁴⁴ [i]