MinT - Best Known (195−98, 195, s)-Nets in Base 2

Best Known (195−98, 195, s)-Nets in Base 2

(195−98, 195, 54)-Net over F₂ — Constructive and digital

Digital (97, 195, 54)-net over F₂, using

t-expansion [i] based on digital (95, 195, 54)-net over F₂, using
- net from sequence [i] based on digital (95, 53)-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 5 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]

(195−98, 195, 65)-Net over F₂ — Digital

Digital (97, 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]

(195−98, 195, 204)-Net over F₂ — Upper bound on s (digital)

There is no digital (97, 195, 205)-net over F₂, because

extracting embedded orthogonal array [i] would yield linear OA(2¹⁹⁵, 205, F₂, 98) (dual of [205, 10, 99]-code), but
- residual code [i] would yield linear OA(2⁹⁷, 106, F₂, 49) (dual of [106, 9, 50]-code), but
  - 1 times truncation [i] would yield linear OA(2⁹⁶, 105, F₂, 48) (dual of [105, 9, 49]-code), but
    - residual code [i] would yield linear OA(2⁴⁸, 56, F₂, 24) (dual of [56, 8, 25]-code), but
      - adding a parity check bit [i] would yield linear OA(2⁴⁹, 57, F₂, 25) (dual of [57, 8, 26]-code), but
        â€œBJVâ€ bound on codes from Brouwerâ€™s database [i]

(195−98, 195, 235)-Net in Base 2 — Upper bound on s

There is no (97, 195, 236)-net in base 2, because

the generalized Rao bound for nets shows that 2^mÂ â‰¥ 53222 451215 293907 762845 953355 055963 675936 537718 842714 290764 > 2¹⁹⁵ [i]