MinT - Best Known (228−123, 228, s)-Nets in Base 4

Best Known (228−123, 228, s)-Nets in Base 4

(228−123, 228, 130)-Net over F₄ — Constructive and digital

Digital (105, 228, 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]

(228−123, 228, 144)-Net over F₄ — Digital

Digital (105, 228, 144)-net over F₄, using

t-expansion [i] based on digital (91, 228, 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]

(228−123, 228, 1316)-Net in Base 4 — Upper bound on s

There is no (105, 228, 1317)-net in base 4, because

1 times m-reduction [i] would yield (105, 227, 1317)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 47180 864664 613583 971914 744607 248565 687333 366375 357919 867728 951601 373969 818198 865684 673483 986610 438957 600507 613891 435465 161051 855162 971520 > 4²²⁷ [i]