MinT - Best Known (245−133, 245, s)-Nets in Base 4

Best Known (245−133, 245, s)-Nets in Base 4

(245−133, 245, 130)-Net over F₄ — Constructive and digital

Digital (112, 245, 130)-net over F₄, using

t-expansion [i] based on digital (105, 245, 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]

(245−133, 245, 165)-Net over F₄ — Digital

Digital (112, 245, 165)-net over F₄, using

t-expansion [i] based on digital (109, 245, 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]

(245−133, 245, 1371)-Net in Base 4 — Upper bound on s

There is no (112, 245, 1372)-net in base 4, because

1 times m-reduction [i] would yield (112, 244, 1372)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 832 513763 760198 024995 288608 361861 542127 343853 416063 443898 872467 916202 826710 030870 424667 517202 181346 829896 505729 958748 808919 578147 659457 983196 473007 > 4²⁴⁴ [i]