MinT - Best Known (146−70, 146, s)-Nets in Base 2

Best Known (146−70, 146, s)-Nets in Base 2

(146−70, 146, 50)-Net over F₂ — Constructive and digital

Digital (76, 146, 50)-net over F₂, using

t-expansion [i] based on digital (75, 146, 50)-net over F₂, using
- net from sequence [i] based on digital (75, 49)-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 1 place 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]

(146−70, 146, 173)-Net over F₂ — Upper bound on s (digital)

There is no digital (76, 146, 174)-net over F₂, because

extracting embedded orthogonal array [i] would yield linear OA(2¹⁴⁶, 174, F₂, 70) (dual of [174, 28, 71]-code), but
- 2 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]

(146−70, 146, 203)-Net in Base 2 — Upper bound on s

There is no (76, 146, 204)-net in base 2, because

the generalized Rao bound for nets shows that 2^mÂ â‰¥ 102 242270 192228 160491 390924 725584 503980 075710 > 2¹⁴⁶ [i]