MinT - Best Known (251−227, 251, s)-Nets in Base 4

Best Known (251−227, 251, s)-Nets in Base 4

(251−227, 251, 34)-Net over F₄ — Constructive and digital

Digital (24, 251, 34)-net over F₄, using

t-expansion [i] based on digital (21, 251, 34)-net over F₄, using
- net from sequence [i] based on digital (21, 33)-sequence over F₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 21 and N(F) ≥ 34, using
    - T₅ from the second tower of function fields by GarcÃa and Stichtenoth over F₄ [i]

(251−227, 251, 35)-Net in Base 4 — Constructive

(24, 251, 35)-net in base 4, using

net from sequence [i] based on (24, 34)-sequence in base 4, using
- base expansion [i] based on digital (48, 34)-sequence over F₂, using
  - Niederreiterâ€“Xing sequence constructionÂ III based on the algebraic function field F/F₂ with g(F)Â =Â 41, N(F)Â =Â 32, 1 place with degree 2, and 2 places with degree 4 [i] based on function field F/F₂ with g(F) = 41 and N(F) ≥ 32, using an explicitly constructive algebraic function field [i]

(251−227, 251, 49)-Net over F₄ — Digital

Digital (24, 251, 49)-net over F₄, using

net from sequence [i] based on digital (24, 48)-sequence over F₄, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 24 and N(F) ≥ 49, using
  - a curve from the manYPoints database [i]

(251−227, 251, 88)-Net in Base 4 — Upper bound on s

There is no (24, 251, 89)-net in base 4, because

77 times m-reduction [i] would yield (24, 174, 89)-net in base 4, but
- extracting embedded OOA [i] would yield OOA(4¹⁷⁴, 89, S₄, 2, 150), but
  - the (dual) Plotkin bound for OOAs shows that MÂ â‰¥ 119261 928031 483718 603762 686555 708520 921421 379436 750618 434354 619648 119989 556601 210019 640964 761726 535700 840448 / 151 > 4¹⁷⁴ [i]