MinT - Best Known (228−115, 228, s)-Nets in Base 4

Best Known (228−115, 228, s)-Nets in Base 4

(228−115, 228, 130)-Net over F₄ — Constructive and digital

Digital (113, 228, 130)-net over F₄, using

t-expansion [i] based on 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]

(228−115, 228, 165)-Net over F₄ — Digital

Digital (113, 228, 165)-net over F₄, using

t-expansion [i] based on digital (109, 228, 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]

(228−115, 228, 1792)-Net in Base 4 — Upper bound on s

There is no (113, 228, 1793)-net in base 4, because

1 times m-reduction [i] would yield (113, 227, 1793)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 47499 566758 694459 205510 757434 876983 022876 016623 363180 898317 222864 488280 219064 965727 094476 990124 366584 153320 351028 481367 450960 458691 863680 > 4²²⁷ [i]