MinT - Best Known (52, 52+180, s)-Nets in Base 2

Best Known (52, 52+180, s)-Nets in Base 2

(52, 52+180, 36)-Net over F₂ — Constructive and digital

Digital (52, 232, 36)-net over F₂, using

t-expansion [i] based on digital (51, 232, 36)-net over F₂, using
- net from sequence [i] based on digital (51, 35)-sequence over F₂, using
  - Niederreiterâ€“Xing sequence constructionÂ III based on the algebraic function field F/F₂ with g(F)Â =Â 41, N(F)Â =Â 32, 1 place with degree 2, and 3 places with degree 4 [i] based on function field F/F₂ with g(F) = 41 and N(F) ≥ 32, using an explicitly constructive algebraic function field [i]

(52, 52+180, 40)-Net over F₂ — Digital

Digital (52, 232, 40)-net over F₂, using

t-expansion [i] based on digital (50, 232, 40)-net over F₂, using
- net from sequence [i] based on digital (50, 39)-sequence over F₂, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂ with g(F) = 50 and N(F) ≥ 40, using
    - Drinfeld modules of rank 1 [i]

(52, 52+180, 67)-Net in Base 2 — Upper bound on s

There is no (52, 232, 68)-net in base 2, because

35 times m-reduction [i] would yield (52, 197, 68)-net in base 2, but
- extracting embedded OOA [i] would yield OOA(2¹⁹⁷, 68, S₂, 3, 145), but
  - the (dual) Plotkin bound for OOAs shows that MÂ â‰¥ 19 283256 531107 883306 503545 108093 951230 266435 925393 514023 616512 / 73 > 2¹⁹⁷ [i]