MinT - Best Known (7, 7+24, s)-Nets in Base 4

Best Known (7, 7+24, s)-Nets in Base 4

(7, 7+24, 21)-Net over F₄ — Constructive and digital

Digital (7, 31, 21)-net over F₄, using

net from sequence [i] based on digital (7, 20)-sequence over F₄, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 7 and N(F) ≥ 21, using
  - an explicitly constructive algebraic function field [i]

(7, 7+24, 34)-Net over F₄ — Upper bound on s (digital)

There is no digital (7, 31, 35)-net over F₄, because

extracting embedded orthogonal array [i] would yield linear OA(4³¹, 35, F₄, 24) (dual of [35, 4, 25]-code), but
- residual code [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, 7+24, 39)-Net in Base 4 — Upper bound on s

There is no (7, 31, 40)-net in base 4, because

extracting embedded orthogonal array [i] would yield OA(4³¹, 40, S₄, 24), but
- the linear programming bound shows that MÂ â‰¥ 75557 863725 914323 419136 / 14875 > 4³¹ [i]