MinT - Best Known (245−137, 245, s)-Nets in Base 4

Best Known (245−137, 245, s)-Nets in Base 4

(245−137, 245, 130)-Net over F₄ — Constructive and digital

Digital (108, 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]

(245−137, 245, 144)-Net over F₄ — Digital

Digital (108, 245, 144)-net over F₄, using

t-expansion [i] based on digital (91, 245, 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]

(245−137, 245, 1205)-Net in Base 4 — Upper bound on s

There is no (108, 245, 1206)-net in base 4, because

1 times m-reduction [i] would yield (108, 244, 1206)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 801 462186 374404 203734 120118 417435 556478 786343 783684 991052 569185 043871 374842 971288 538242 107282 107462 046558 335954 138747 771169 719233 380390 976440 766170 > 4²⁴⁴ [i]