MinT - Best Known (118−55, 118, s)-Nets in Base 2

Best Known (118−55, 118, s)-Nets in Base 2

(118−55, 118, 43)-Net over F₂ — Constructive and digital

Digital (63, 118, 43)-net over F₂, using

t-expansion [i] based on digital (59, 118, 43)-net over F₂, using
- net from sequence [i] based on digital (59, 42)-sequence over F₂, using
  - Niederreiterâ€“Xing sequence constructionÂ III based on the algebraic function field F/F₂ with g(F)Â =Â 54, N(F)Â =Â 42, and 1 place with degree 6 [i] based on function field F/F₂ with g(F) = 54 and N(F) ≥ 42, using an explicitly constructive algebraic function field [i]

(118−55, 118, 44)-Net over F₂ — Digital

Digital (63, 118, 44)-net over F₂, using

t-expansion [i] based on digital (62, 118, 44)-net over F₂, using
- net from sequence [i] based on digital (62, 43)-sequence over F₂, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂ with g(F) = 62 and N(F) ≥ 44, using
    - a function field from a table by Niederreiter and Xing [i]

(118−55, 118, 167)-Net over F₂ — Upper bound on s (digital)

There is no digital (63, 118, 168)-net over F₂, because

1 times m-reduction [i] would yield digital (63, 117, 168)-net over F₂, but
- extracting embedded orthogonal array [i] would yield linear OA(2¹¹⁷, 168, F₂, 54) (dual of [168, 51, 55]-code), but
  - constructionÂ Y1 [i] would yield
    1. OA(2¹¹⁶, 148, S₂, 54), but
      - the linear programming bound shows that MÂ â‰¥ 67 089316 460630 621624 223128 238837 181334 945792 / 674 571975 > 2¹¹⁶ [i]
    2. OA(2⁵¹, 168, S₂, 20), but
      - discarding factors would yield OA(2⁵¹, 159, S₂, 20), but
        the Rao or (dual) Hamming bound shows that MÂ â‰¥ 2282 987168 569557 > 2⁵¹ [i]

(118−55, 118, 183)-Net in Base 2 — Upper bound on s

There is no (63, 118, 184)-net in base 2, because

1 times m-reduction [i] would yield (63, 117, 184)-net in base 2, but
- the generalized Rao bound for nets shows that 2^mÂ â‰¥ 182467 222295 912684 213142 696981 078356 > 2¹¹⁷ [i]