MinT - Best Known (252−229, 252, s)-Nets in Base 4

Best Known (252−229, 252, s)-Nets in Base 4

(252−229, 252, 34)-Net over F₄ — Constructive and digital

Digital (23, 252, 34)-net over F₄, using

t-expansion [i] based on digital (21, 252, 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]

(252−229, 252, 45)-Net over F₄ — Digital

Digital (23, 252, 45)-net over F₄, using

net from sequence [i] based on digital (23, 44)-sequence over F₄, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 23 and N(F) ≥ 45, using
  - a curve from the manYPoints database [i]

(252−229, 252, 81)-Net in Base 4 — Upper bound on s

There is no (23, 252, 82)-net in base 4, because

10 times m-reduction [i] would yield (23, 242, 82)-net in base 4, but
- extracting embedded OOA [i] would yield OOA(4²⁴², 82, S₄, 3, 219), but
  - the (dual) Plotkin bound for OOAs shows that MÂ â‰¥ 2896 982654 693240 790722 122229 248058 331734 699076 393093 227997 297658 532344 973624 742495 471644 063642 920549 236319 392080 665346 046975 721430 379528 764237 283328 / 55 > 4²⁴² [i]