MinT - Best Known (137−126, 137, s)-Nets in Base 4

Best Known (137−126, 137, s)-Nets in Base 4

(137−126, 137, 27)-Net over F₄ — Constructive and digital

Digital (11, 137, 27)-net over F₄, using

t-expansion [i] based on digital (10, 137, 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]

(137−126, 137, 44)-Net in Base 4 — Upper bound on s

There is no (11, 137, 45)-net in base 4, because

6 times m-reduction [i] would yield (11, 131, 45)-net in base 4, but
- extracting embedded OOA [i] would yield OOA(4¹³¹, 45, S₄, 3, 120), but
  - the (dual) Plotkin bound for OOAs shows that MÂ â‰¥ 1452 495967 392894 355393 274435 948981 116111 418687 645795 235310 955933 795262 298203 357184 / 121 > 4¹³¹ [i]