MinT - Best Known (228−139, 228, s)-Nets in Base 4

Best Known (228−139, 228, s)-Nets in Base 4

(228−139, 228, 104)-Net over F₄ — Constructive and digital

Digital (89, 228, 104)-net over F₄, using

t-expansion [i] based on digital (73, 228, 104)-net over F₄, using
- net from sequence [i] based on digital (73, 103)-sequence over F₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 73 and N(F) ≥ 104, using
    - F₆ from the tower of function fields by GarcÃa and Stichtenoth over F₄ [i]

(228−139, 228, 129)-Net over F₄ — Digital

Digital (89, 228, 129)-net over F₄, using

t-expansion [i] based on digital (81, 228, 129)-net over F₄, using
- net from sequence [i] based on digital (81, 128)-sequence over F₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 81 and N(F) ≥ 129, using
    - Drinfeld modules of rank 1 [i]

(228−139, 228, 790)-Net in Base 4 — Upper bound on s

There is no (89, 228, 791)-net in base 4, because

1 times m-reduction [i] would yield (89, 227, 791)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 49811 324766 130235 543482 550271 954738 147406 460169 177965 053451 451839 178835 378632 326996 969524 934235 169130 911196 065381 437256 676607 118233 681880 > 4²²⁷ [i]