MinT - Best Known (15−13, 15, s)-Nets in Base 5

Best Known (15−13, 15, s)-Nets in Base 5

(15−13, 15, 12)-Net over F₅ — Constructive and digital

Digital (2, 15, 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]

(15−13, 15, 16)-Net over F₅ — Upper bound on s (digital)

There is no digital (2, 15, 17)-net over F₅, because

3 times m-reduction [i] would yield digital (2, 12, 17)-net over F₅, but
- extracting embedded orthogonal array [i] would yield linear OA(5¹², 17, F₅, 10) (dual of [17, 5, 11]-code), but
  - constructionÂ Y1 [i] would yield
    1. OA(5¹¹, 13, S₅, 10), but
      - the (dual) Plotkin bound shows that MÂ â‰¥ 732 421875 / 11 > 5¹¹ [i]
    2. linear OA(5⁵, 17, F₅, 4) (dual of [17, 12, 5]-code), but
      - discarding factors / shortening the dual code would yield linear OA(5⁵, 13, F₅, 4) (dual of [13, 8, 5]-code), but
        â€œBouâ€ bound on codes from Brouwerâ€™s database [i]

(15−13, 15, 20)-Net in Base 5 — Upper bound on s

There is no (2, 15, 21)-net in base 5, because

extracting embedded OOA [i] would yield OOA(5¹⁵, 21, S₅, 2, 13), but
- the linear programming bound for OOAs shows that MÂ â‰¥ 370828 391587 951768 035888 671875 / 11 198228 218350 722451 > 5¹⁵ [i]