MinT - Best Known (249−141, 249, s)-Nets in Base 4

Best Known (249−141, 249, s)-Nets in Base 4

(249−141, 249, 130)-Net over F₄ — Constructive and digital

Digital (108, 249, 130)-net over F₄, using

t-expansion [i] based on digital (105, 249, 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]

(249−141, 249, 144)-Net over F₄ — Digital

Digital (108, 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]

(249−141, 249, 1161)-Net in Base 4 — Upper bound on s

There is no (108, 249, 1162)-net in base 4, because

1 times m-reduction [i] would yield (108, 248, 1162)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 216588 956334 912294 837723 550232 046543 391542 293392 097955 286956 503569 701420 100629 499766 436751 913700 308236 174568 741372 587843 243346 266545 885745 631873 632600 > 4²⁴⁸ [i]