MinT - Best Known (7−6, 7, s)-Nets in Base 4

Best Known (7−6, 7, s)-Nets in Base 4

(7−6, 7, 9)-Net over F₄ — Constructive and digital

Digital (1, 7, 9)-net over F₄, using

net from sequence [i] based on digital (1, 8)-sequence over F₄, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 1 and N(F) ≥ 9, using
  - the Hermitian function field over F₄ [i]

(7−6, 7, 9)-Net over F₄ — Upper bound on s (digital)

There is no digital (1, 7, 10)-net over F₄, because

extracting embedded orthogonal array [i] would yield linear OA(4⁷, 10, F₄, 6) (dual of [10, 3, 7]-code), but
- bound for almost-MDS-codes with dimension nÂ =Â 3 [i]

(7−6, 7, 12)-Net in Base 4 — Upper bound on s

There is no (1, 7, 13)-net in base 4, because

1 times m-reduction [i] would yield (1, 6, 13)-net in base 4, but
- extracting embedded OOA [i] would yield OOA(4⁶, 13, S₄, 2, 5), but
  - the linear programming bound for OOAs shows that MÂ â‰¥ 225280 / 53 > 4⁶ [i]