MinT - Best Known (257−163, 257, s)-Nets in Base 4

Best Known (257−163, 257, s)-Nets in Base 4

(257−163, 257, 104)-Net over F₄ — Constructive and digital

Digital (94, 257, 104)-net over F₄, using

t-expansion [i] based on digital (73, 257, 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]

(257−163, 257, 144)-Net over F₄ — Digital

Digital (94, 257, 144)-net over F₄, using

t-expansion [i] based on digital (91, 257, 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]

(257−163, 257, 760)-Net in Base 4 — Upper bound on s

There is no (94, 257, 761)-net in base 4, because

1 times m-reduction [i] would yield (94, 256, 761)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 14437 488707 823987 285181 470704 857375 720036 332745 185231 186071 287768 501102 950560 745072 323635 418324 825058 036490 849538 138290 219894 753716 007969 073477 384118 134140 > 4²⁵⁶ [i]