MinT - Best Known (226−113, 226, s)-Nets in Base 4

Best Known (226−113, 226, s)-Nets in Base 4

(226−113, 226, 130)-Net over F₄ — Constructive and digital

Digital (113, 226, 130)-net over F₄, using

t-expansion [i] based on digital (105, 226, 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]

(226−113, 226, 165)-Net over F₄ — Digital

Digital (113, 226, 165)-net over F₄, using

t-expansion [i] based on digital (109, 226, 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]

(226−113, 226, 1853)-Net in Base 4 — Upper bound on s

There is no (113, 226, 1854)-net in base 4, because

1 times m-reduction [i] would yield (113, 225, 1854)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 2977 389487 827962 216177 822000 472071 173365 179008 465804 299137 287151 794532 164224 602310 923210 794567 483389 297834 918949 259261 242021 292727 895267 > 4²²⁵ [i]