MinT - Best Known (97, 192, s)-Nets in Base 2

Best Known (97, 192, s)-Nets in Base 2

(97, 192, 54)-Net over F₂ — Constructive and digital

Digital (97, 192, 54)-net over F₂, using

t-expansion [i] based on digital (95, 192, 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]

(97, 192, 65)-Net over F₂ — Digital

Digital (97, 192, 65)-net over F₂, using

t-expansion [i] based on digital (95, 192, 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]

(97, 192, 213)-Net over F₂ — Upper bound on s (digital)

There is no digital (97, 192, 214)-net over F₂, because

1 times m-reduction [i] would yield digital (97, 191, 214)-net over F₂, but
- extracting embedded orthogonal array [i] would yield linear OA(2¹⁹¹, 214, F₂, 94) (dual of [214, 23, 95]-code), but
  - residual code [i] would yield OA(2⁹⁷, 119, S₂, 47), but
    - 1 times truncation [i] would yield OA(2⁹⁶, 118, S₂, 46), but
      - the linear programming bound shows that MÂ â‰¥ 3649 328393 569529 653896 227896 426496 / 35409 > 2⁹⁶ [i]

(97, 192, 243)-Net in Base 2 — Upper bound on s

There is no (97, 192, 244)-net in base 2, because

1 times m-reduction [i] would yield (97, 191, 244)-net in base 2, but
- the generalized Rao bound for nets shows that 2^mÂ â‰¥ 3286 345835 685558 350605 490942 186974 822130 251388 005033 010828 > 2¹⁹¹ [i]