MinT - Best Known (7, 7+68, s)-Nets in Base 4

Best Known (7, 7+68, s)-Nets in Base 4

(7, 7+68, 21)-Net over F₄ — Constructive and digital

Digital (7, 75, 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+68, 31)-Net in Base 4 — Upper bound on s

There is no (7, 75, 32)-net in base 4, because

14 times m-reduction [i] would yield (7, 61, 32)-net in base 4, but
- extracting embedded OOA [i] would yield OOA(4⁶¹, 32, S₄, 2, 54), but
  - the (dual) Plotkin bound for OOAs shows that MÂ â‰¥ 340 282366 920938 463463 374607 431768 211456 / 55 > 4⁶¹ [i]