MinT - Best Known (257−133, 257, s)-Nets in Base 4

Best Known (257−133, 257, s)-Nets in Base 4

(257−133, 257, 130)-Net over F₄ — Constructive and digital

Digital (124, 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]

(257−133, 257, 168)-Net over F₄ — Digital

Digital (124, 257, 168)-net over F₄, using

t-expansion [i] based on digital (115, 257, 168)-net over F₄, using
- net from sequence [i] based on digital (115, 167)-sequence over F₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 115 and N(F) ≥ 168, using
    - Drinfeld modules of rank 1 [i]

(257−133, 257, 1779)-Net in Base 4 — Upper bound on s

There is no (124, 257, 1780)-net in base 4, because

1 times m-reduction [i] would yield (124, 256, 1780)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 13701 907319 066185 777715 999321 277417 387929 470381 123033 500097 592338 537704 191286 285201 298303 680641 216011 872092 452660 989824 667492 213316 632105 185485 149723 443815 > 4²⁵⁶ [i]