MinT - Best Known (81−72, 81, s)-Nets in Base 5

Best Known (81−72, 81, s)-Nets in Base 5

(81−72, 81, 26)-Net over F₅ — Constructive and digital

Digital (9, 81, 26)-net over F₅, using

net from sequence [i] based on digital (9, 25)-sequence over F₅, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₅ with g(F) = 9 and N(F) ≥ 26, using
  - an explicitly constructive algebraic function field [i]

(81−72, 81, 52)-Net over F₅ — Upper bound on s (digital)

There is no digital (9, 81, 53)-net over F₅, because

32 times m-reduction [i] would yield digital (9, 49, 53)-net over F₅, but
- extracting embedded orthogonal array [i] would yield linear OA(5⁴⁹, 53, F₅, 40) (dual of [53, 4, 41]-code), but
  - residual code [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]

(81−72, 81, 55)-Net in Base 5 — Upper bound on s

There is no (9, 81, 56)-net in base 5, because

33 times m-reduction [i] would yield (9, 48, 56)-net in base 5, but
- extracting embedded orthogonal array [i] would yield OA(5⁴⁸, 56, S₅, 39), but
  - the linear programming bound shows that MÂ â‰¥ 877520 278663 723729 550838 470458 984375 / 207 > 5⁴⁸ [i]