MinT - Best Known (215−106, 215, s)-Nets in Base 2

Best Known (215−106, 215, s)-Nets in Base 2

(215−106, 215, 56)-Net over F₂ — Constructive and digital

Digital (109, 215, 56)-net over F₂, using

t-expansion [i] based on digital (105, 215, 56)-net over F₂, using
- net from sequence [i] based on digital (105, 55)-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 7 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]

(215−106, 215, 65)-Net over F₂ — Digital

Digital (109, 215, 65)-net over F₂, using

t-expansion [i] based on digital (95, 215, 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]

(215−106, 215, 230)-Net in Base 2 — Upper bound on s

There is no (109, 215, 231)-net in base 2, because

4 times m-reduction [i] would yield (109, 211, 231)-net in base 2, but
- extracting embedded orthogonal array [i] would yield OA(2²¹¹, 231, S₂, 102), but
  - the linear programming bound shows that MÂ â‰¥ 458 319093 341560 876509 339256 479333 721451 831455 183189 732178 761374 236672 / 109681 > 2²¹¹ [i]