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

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

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

Digital (111, 222, 130)-net over F₄, using

t-expansion [i] based on digital (105, 222, 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+111, 165)-Net over F₄ — Digital

Digital (111, 222, 165)-net over F₄, using

t-expansion [i] based on digital (109, 222, 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+111, 1822)-Net in Base 4 — Upper bound on s

There is no (111, 222, 1823)-net in base 4, because

1 times m-reduction [i] would yield (111, 221, 1823)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 11 552275 122436 082858 087411 823395 797920 673277 879317 626139 069508 829011 874939 966973 418412 631400 279219 360650 910500 232384 506368 641635 960288 > 4²²¹ [i]