MinT - Best Known (111, 111+123, s)-Nets in Base 4

Best Known (111, 111+123, s)-Nets in Base 4

(111, 111+123, 130)-Net over F₄ — Constructive and digital

Digital (111, 234, 130)-net over F₄, using

t-expansion [i] based on digital (105, 234, 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]

(111, 111+123, 165)-Net over F₄ — Digital

Digital (111, 234, 165)-net over F₄, using

t-expansion [i] based on digital (109, 234, 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]

(111, 111+123, 1516)-Net in Base 4 — Upper bound on s

There is no (111, 234, 1517)-net in base 4, because

1 times m-reduction [i] would yield (111, 233, 1517)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 196 546211 046105 847668 896579 219190 986319 188580 209336 155571 922438 599147 307587 869974 427488 905682 557125 676112 710297 767454 785772 619270 706426 326640 > 4²³³ [i]