MinT - Best Known (187−90, 187, s)-Nets in Base 2

Best Known (187−90, 187, s)-Nets in Base 2

(187−90, 187, 54)-Net over F₂ — Constructive and digital

Digital (97, 187, 54)-net over F₂, using

t-expansion [i] based on digital (95, 187, 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]

(187−90, 187, 65)-Net over F₂ — Digital

Digital (97, 187, 65)-net over F₂, using

t-expansion [i] based on digital (95, 187, 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]

(187−90, 187, 224)-Net over F₂ — Upper bound on s (digital)

There is no digital (97, 187, 225)-net over F₂, because

4 times m-reduction [i] would yield digital (97, 183, 225)-net over F₂, but
- extracting embedded orthogonal array [i] would yield linear OA(2¹⁸³, 225, F₂, 86) (dual of [225, 42, 87]-code), but
  - constructionÂ Y1 [i] would yield
    1. 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]
    2. OA(2⁴², 225, S₂, 14), but
      - discarding factors would yield OA(2⁴², 219, S₂, 14), but
        the Rao or (dual) Hamming bound shows that MÂ â‰¥ 4 498367 189624 > 2⁴² [i]

(187−90, 187, 252)-Net in Base 2 — Upper bound on s

There is no (97, 187, 253)-net in base 2, because

the generalized Rao bound for nets shows that 2^mÂ â‰¥ 196 244562 658909 562870 503757 745607 746275 367643 768694 038102 > 2¹⁸⁷ [i]