MinT - Best Known (7, 7+30, s)-Nets in Base 5

Best Known (7, 7+30, s)-Nets in Base 5

(7, 7+30, 22)-Net over F₅ — Constructive and digital

Digital (7, 37, 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, 7+30, 42)-Net over F₅ — Upper bound on s (digital)

There is no digital (7, 37, 43)-net over F₅, because

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, 7+30, 49)-Net in Base 5 — Upper bound on s

There is no (7, 37, 50)-net in base 5, because

extracting embedded orthogonal array [i] would yield OA(5³⁷, 50, S₅, 30), but
- the linear programming bound shows that MÂ â‰¥ 1901 298674 056306 481361 389160 156250 / 18 732649 > 5³⁷ [i]