MinT - Best Known (257−145, 257, s)-Nets in Base 4

Best Known (257−145, 257, s)-Nets in Base 4

(257−145, 257, 130)-Net over F₄ — Constructive and digital

Digital (112, 257, 130)-net over F₄, using

t-expansion [i] based on digital (105, 257, 130)-net over F₄, using
- net from sequence [i] based on digital (105, 129)-sequence over F₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 105 and N(F) ≥ 130, using
    - T₇ from the second tower of function fields by GarcÃa and Stichtenoth over F₄ [i]

(257−145, 257, 165)-Net over F₄ — Digital

Digital (112, 257, 165)-net over F₄, using

t-expansion [i] based on digital (109, 257, 165)-net over F₄, using
- net from sequence [i] based on digital (109, 164)-sequence over F₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 109 and N(F) ≥ 165, using
    - Drinfeld modules of rank 1 [i]

(257−145, 257, 1215)-Net in Base 4 — Upper bound on s

There is no (112, 257, 1216)-net in base 4, because

1 times m-reduction [i] would yield (112, 256, 1216)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 14104 240177 561808 645971 318714 803265 023596 065726 035944 188554 055827 368853 989559 327200 707454 047822 080344 552568 954682 861163 485680 843370 950389 167468 781109 868810 > 4²⁵⁶ [i]