MinT - Best Known (15−12, 15, s)-Nets in Base 5

Best Known (15−12, 15, s)-Nets in Base 5

(15−12, 15, 16)-Net over F₅ — Constructive and digital

Digital (3, 15, 16)-net over F₅, using

net from sequence [i] based on digital (3, 15)-sequence over F₅, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₅ with g(F) = 3 and N(F) ≥ 16, using
  - an explicitly constructive algebraic function field [i]

(15−12, 15, 24)-Net over F₅ — Upper bound on s (digital)

There is no digital (3, 15, 25)-net over F₅, because

extracting embedded orthogonal array [i] would yield linear OA(5¹⁵, 25, F₅, 12) (dual of [25, 10, 13]-code), but
- constructionÂ Y1 [i] would yield
  1. linear OA(5¹⁴, 17, F₅, 12) (dual of [17, 3, 13]-code), but
    - a result by Hill [i]
  2. linear OA(5¹⁰, 25, F₅, 8) (dual of [25, 15, 9]-code), but
    - discarding factors / shortening the dual code would yield linear OA(5¹⁰, 22, F₅, 8) (dual of [22, 12, 9]-code), but
      - constructionÂ Y1 [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]
        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]

(15−12, 15, 37)-Net in Base 5 — Upper bound on s

There is no (3, 15, 38)-net in base 5, because

the generalized Rao bound for nets shows that 5^mÂ â‰¥ 30599 281729 > 5¹⁵ [i]