MinT - Best Known (172−162, 172, s)-Nets in Base 4

Best Known (172−162, 172, s)-Nets in Base 4

(172−162, 172, 27)-Net over F₄ — Constructive and digital

Digital (10, 172, 27)-net over F₄, using

net from sequence [i] based on digital (10, 26)-sequence over F₄, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 10 and N(F) ≥ 27, using
  - an explicitly constructive algebraic function field [i]

(172−162, 172, 40)-Net in Base 4 — Upper bound on s

There is no (10, 172, 41)-net in base 4, because

53 times m-reduction [i] would yield (10, 119, 41)-net in base 4, but
- extracting embedded OOA [i] would yield OOA(4¹¹⁹, 41, S₄, 3, 109), but
  - the (dual) Plotkin bound for OOAs shows that MÂ â‰¥ 25 619282 439286 572778 957813 760772 318479 498516 504696 474892 763289 918742 986752 / 55 > 4¹¹⁹ [i]