MinT - Best Known (11−9, 11, s)-Nets in Base 5

Best Known (11−9, 11, s)-Nets in Base 5

(11−9, 11, 12)-Net over F₅ — Constructive and digital

Digital (2, 11, 12)-net over F₅, using

net from sequence [i] based on digital (2, 11)-sequence over F₅, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₅ with g(F) = 2 and N(F) ≥ 12, using
  - an explicitly constructive algebraic function field [i]

(11−9, 11, 21)-Net over F₅ — Upper bound on s (digital)

There is no digital (2, 11, 22)-net over F₅, because

1 times m-reduction [i] would yield digital (2, 10, 22)-net over F₅, but
- extracting embedded orthogonal array [i] would yield linear OA(5¹⁰, 22, F₅, 8) (dual of [22, 12, 9]-code), but
  - constructionÂ Y1 [i] would yield
    1. 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. linear OA(5¹², 22, F₅, 10) (dual of [22, 10, 11]-code), but
      - discarding factors / shortening the dual code would yield linear OA(5¹², 18, F₅, 10) (dual of [18, 6, 11]-code), but
        residual code [i] would yield OA(5², 7, S₅, 2), but
        bound for OAs with strength kÂ =Â 2 [i]
        the Rao or (dual) Hamming bound shows that MÂ â‰¥ 29 > 5² [i]

(11−9, 11, 27)-Net in Base 5 — Upper bound on s

There is no (2, 11, 28)-net in base 5, because

1 times m-reduction [i] would yield (2, 10, 28)-net in base 5, but
- extracting embedded OOA [i] would yield OOA(5¹⁰, 28, S₅, 2, 8), but
  - the linear programming bound for OOAs shows that MÂ â‰¥ 2 898660 763940 468750 / 275216 094827 > 5¹⁰ [i]