MinT - Best Known (159−82, 159, s)-Nets in Base 2

Best Known (159−82, 159, s)-Nets in Base 2

(159−82, 159, 50)-Net over F₂ — Constructive and digital

Digital (77, 159, 50)-net over F₂, using

t-expansion [i] based on digital (75, 159, 50)-net over F₂, using
- net from sequence [i] based on digital (75, 49)-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 1 place 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]

(159−82, 159, 52)-Net over F₂ — Digital

Digital (77, 159, 52)-net over F₂, using

net from sequence [i] based on digital (77, 51)-sequence over F₂, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂ with g(F) = 77 and N(F) ≥ 52, using
  - Drinfeld modules of rank 1 [i]

(159−82, 159, 163)-Net over F₂ — Upper bound on s (digital)

There is no digital (77, 159, 164)-net over F₂, because

2 times m-reduction [i] would yield digital (77, 157, 164)-net over F₂, but
- extracting embedded orthogonal array [i] would yield linear OA(2¹⁵⁷, 164, F₂, 80) (dual of [164, 7, 81]-code), but
  - Griesmer bound [i]

(159−82, 159, 182)-Net in Base 2 — Upper bound on s

There is no (77, 159, 183)-net in base 2, because

the generalized Rao bound for nets shows that 2^mÂ â‰¥ 782716 143120 767003 361125 383967 809086 779319 095680 > 2¹⁵⁹ [i]