MinT - Best Known (195−92, 195, s)-Nets in Base 2

Best Known (195−92, 195, s)-Nets in Base 2

(195−92, 195, 55)-Net over F₂ — Constructive and digital

Digital (103, 195, 55)-net over F₂, using

t-expansion [i] based on digital (100, 195, 55)-net over F₂, using
- net from sequence [i] based on digital (100, 54)-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 6 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]

(195−92, 195, 65)-Net over F₂ — Digital

Digital (103, 195, 65)-net over F₂, using

t-expansion [i] based on digital (95, 195, 65)-net over F₂, using
- net from sequence [i] based on digital (95, 64)-sequence over F₂, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂ with g(F) = 95 and N(F) ≥ 65, using
    - Drinfeld modules of rank 1 [i]

(195−92, 195, 231)-Net over F₂ — Upper bound on s (digital)

There is no digital (103, 195, 232)-net over F₂, because

extracting embedded orthogonal array [i] would yield linear OA(2¹⁹⁵, 232, F₂, 92) (dual of [232, 37, 93]-code), but
- constructionÂ Y1 [i] would yield
  1. linear OA(2¹⁹⁴, 220, F₂, 92) (dual of [220, 26, 93]-code), but
    - adding a parity check bit [i] would yield linear OA(2¹⁹⁵, 221, F₂, 93) (dual of [221, 26, 94]-code), but
      - â€œBKâ€ bound on codes from Brouwerâ€™s database [i]
  2. OA(2³⁷, 232, S₂, 12), but
    - discarding factors would yield OA(2³⁷, 217, S₂, 12), but
      - the Rao or (dual) Hamming bound shows that MÂ â‰¥ 139173 045698 > 2³⁷ [i]

(195−92, 195, 277)-Net in Base 2 — Upper bound on s

There is no (103, 195, 278)-net in base 2, because

the generalized Rao bound for nets shows that 2^mÂ â‰¥ 56973 645297 153357 193578 927937 478842 020442 779226 381548 376512 > 2¹⁹⁵ [i]