MinT - Best Known (191−98, 191, s)-Nets in Base 2

Best Known (191−98, 191, s)-Nets in Base 2

(191−98, 191, 53)-Net over F₂ — Constructive and digital

Digital (93, 191, 53)-net over F₂, using

t-expansion [i] based on digital (90, 191, 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]

(191−98, 191, 60)-Net over F₂ — Digital

Digital (93, 191, 60)-net over F₂, using

t-expansion [i] based on digital (92, 191, 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]

(191−98, 191, 195)-Net over F₂ — Upper bound on s (digital)

There is no digital (93, 191, 196)-net over F₂, because

2 times m-reduction [i] would yield digital (93, 189, 196)-net over F₂, but
- extracting embedded orthogonal array [i] would yield linear OA(2¹⁸⁹, 196, F₂, 96) (dual of [196, 7, 97]-code), but
  - Griesmer bound [i]

(191−98, 191, 219)-Net in Base 2 — Upper bound on s

There is no (93, 191, 220)-net in base 2, because

the generalized Rao bound for nets shows that 2^mÂ â‰¥ 3468 899262 731309 080755 782955 179603 847913 575183 919933 398570 > 2¹⁹¹ [i]