MinT - Best Known (7, 57, s)-Nets in Base 5

Best Known (7, 57, s)-Nets in Base 5

(7, 57, 22)-Net over F₅ — Constructive and digital

Digital (7, 57, 22)-net over F₅, using

net from sequence [i] based on digital (7, 21)-sequence over F₅, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₅ with g(F) = 7 and N(F) ≥ 22, using
  - an explicitly constructive algebraic function field [i]

(7, 57, 42)-Net over F₅ — Upper bound on s (digital)

There is no digital (7, 57, 43)-net over F₅, because

20 times m-reduction [i] would yield digital (7, 37, 43)-net over F₅, but
- extracting embedded orthogonal array [i] would yield linear OA(5³⁷, 43, F₅, 30) (dual of [43, 6, 31]-code), but
  - residual code [i] would yield linear OA(5⁷, 12, F₅, 6) (dual of [12, 5, 7]-code), but
    - a result by Dodunekov and Landjev [i]

(7, 57, 45)-Net in Base 5 — Upper bound on s

There is no (7, 57, 46)-net in base 5, because

19 times m-reduction [i] would yield (7, 38, 46)-net in base 5, but
- extracting embedded orthogonal array [i] would yield OA(5³⁸, 46, S₅, 31), but
  - the linear programming bound shows that MÂ â‰¥ 4 092726 157978 177070 617675 781250 / 10619 > 5³⁸ [i]