MinT - Best Known (249−157, 249, s)-Nets in Base 4

Best Known (249−157, 249, s)-Nets in Base 4

(249−157, 249, 104)-Net over F₄ — Constructive and digital

Digital (92, 249, 104)-net over F₄, using

t-expansion [i] based on digital (73, 249, 104)-net over F₄, using
- net from sequence [i] based on digital (73, 103)-sequence over F₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 73 and N(F) ≥ 104, using
    - F₆ from the tower of function fields by GarcÃa and Stichtenoth over F₄ [i]

(249−157, 249, 144)-Net over F₄ — Digital

Digital (92, 249, 144)-net over F₄, using

t-expansion [i] based on digital (91, 249, 144)-net over F₄, using
- net from sequence [i] based on digital (91, 143)-sequence over F₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 91 and N(F) ≥ 144, using
    - Drinfeld modules of rank 1 [i]

(249−157, 249, 754)-Net in Base 4 — Upper bound on s

There is no (92, 249, 755)-net in base 4, because

1 times m-reduction [i] would yield (92, 248, 755)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 219345 483466 752005 529854 623267 288473 855221 659203 237157 034800 089685 696931 272685 423379 184986 873326 244101 161500 614585 962710 192478 142140 809055 612276 046524 > 4²⁴⁸ [i]