MinT - Best Known (56−45, 56, s)-Nets in Base 5

Best Known (56−45, 56, s)-Nets in Base 5

(56−45, 56, 32)-Net over F₅ — Constructive and digital

Digital (11, 56, 32)-net over F₅, using

net from sequence [i] based on digital (11, 31)-sequence over F₅, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₅ with g(F) = 11 and N(F) ≥ 32, using
  - an explicitly constructive algebraic function field [i]

(56−45, 56, 64)-Net over F₅ — Upper bound on s (digital)

There is no digital (11, 56, 65)-net over F₅, because

extracting embedded orthogonal array [i] would yield linear OA(5⁵⁶, 65, F₅, 45) (dual of [65, 9, 46]-code), but
- constructionÂ Y1 [i] would yield
  1. linear OA(5⁵⁵, 59, F₅, 45) (dual of [59, 4, 46]-code), but
    - residual code [i] would yield linear OA(5¹⁰, 13, F₅, 9) (dual of [13, 3, 10]-code), but
      - 1 times truncation [i] would yield linear OA(5⁹, 12, F₅, 8) (dual of [12, 3, 9]-code), but
        bound for almost-MDS-codes with dimension nÂ =Â 3 [i]
  2. OA(5⁹, 65, S₅, 6), but
    - discarding factors would yield OA(5⁹, 58, S₅, 6), but
      - the Rao or (dual) Hamming bound shows that MÂ â‰¥ 2 001465 > 5⁹ [i]

(56−45, 56, 65)-Net in Base 5 — Upper bound on s

There is no (11, 56, 66)-net in base 5, because

1 times m-reduction [i] would yield (11, 55, 66)-net in base 5, but
- extracting embedded orthogonal array [i] would yield OA(5⁵⁵, 66, S₅, 44), but
  - the linear programming bound shows that MÂ â‰¥ 975608 482889 356309 897266 328334 808349 609375 / 3111 > 5⁵⁵ [i]