MinT - Best Known (189−93, 189, s)-Nets in Base 2

Best Known (189−93, 189, s)-Nets in Base 2

(189−93, 189, 54)-Net over F₂ — Constructive and digital

Digital (96, 189, 54)-net over F₂, using

t-expansion [i] based on digital (95, 189, 54)-net over F₂, using
- net from sequence [i] based on digital (95, 53)-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 5 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]

(189−93, 189, 65)-Net over F₂ — Digital

Digital (96, 189, 65)-net over F₂, using

t-expansion [i] based on digital (95, 189, 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]

(189−93, 189, 210)-Net over F₂ — Upper bound on s (digital)

There is no digital (96, 189, 211)-net over F₂, because

7 times m-reduction [i] would yield digital (96, 182, 211)-net over F₂, but
- extracting embedded orthogonal array [i] would yield linear OA(2¹⁸², 211, F₂, 86) (dual of [211, 29, 87]-code), but
  - adding a parity check bit [i] would yield linear OA(2¹⁸³, 212, F₂, 87) (dual of [212, 29, 88]-code), but
    - â€œBKâ€ bound on codes from Brouwerâ€™s database [i]

(189−93, 189, 243)-Net in Base 2 — Upper bound on s

There is no (96, 189, 244)-net in base 2, because

1 times m-reduction [i] would yield (96, 188, 244)-net in base 2, but
- the generalized Rao bound for nets shows that 2^mÂ â‰¥ 415 736221 639699 618112 478508 223806 296942 032358 730330 579420 > 2¹⁸⁸ [i]