MinT - Best Known (182−90, 182, s)-Nets in Base 2

Best Known (182−90, 182, s)-Nets in Base 2

(182−90, 182, 53)-Net over F₂ — Constructive and digital

Digital (92, 182, 53)-net over F₂, using

t-expansion [i] based on digital (90, 182, 53)-net over F₂, using
- net from sequence [i] based on digital (90, 52)-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 4 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]

(182−90, 182, 60)-Net over F₂ — Digital

Digital (92, 182, 60)-net over F₂, using

net from sequence [i] based on digital (92, 59)-sequence over F₂, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂ with g(F) = 92 and N(F) ≥ 60, using
  - Drinfeld modules of rank 1 [i]

(182−90, 182, 202)-Net over F₂ — Upper bound on s (digital)

There is no digital (92, 182, 203)-net over F₂, because

2 times m-reduction [i] would yield digital (92, 180, 203)-net over F₂, but
- extracting embedded orthogonal array [i] would yield linear OA(2¹⁸⁰, 203, F₂, 88) (dual of [203, 23, 89]-code), but
  - residual code [i] would yield OA(2⁹², 114, S₂, 44), but
    - the linear programming bound shows that MÂ â‰¥ 26 699890 767307 081769 024311 263232 / 4557 > 2⁹² [i]

(182−90, 182, 230)-Net in Base 2 — Upper bound on s

There is no (92, 182, 231)-net in base 2, because

the generalized Rao bound for nets shows that 2^mÂ â‰¥ 7 030366 636848 093456 042903 577982 643900 812926 459827 559328 > 2¹⁸² [i]