MinT - Best Known (111, 111+125, s)-Nets in Base 4

Best Known (111, 111+125, s)-Nets in Base 4

(111, 111+125, 130)-Net over F₄ — Constructive and digital

Digital (111, 236, 130)-net over F₄, using

t-expansion [i] based on digital (105, 236, 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+125, 165)-Net over F₄ — Digital

Digital (111, 236, 165)-net over F₄, using

t-expansion [i] based on digital (109, 236, 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+125, 1476)-Net in Base 4 — Upper bound on s

There is no (111, 236, 1477)-net in base 4, because

1 times m-reduction [i] would yield (111, 235, 1477)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 3096 472301 509221 453250 454386 502596 030905 424551 202692 950727 566594 581427 008215 756458 033775 370729 881833 028920 464817 719835 565282 799833 583927 732960 > 4²³⁵ [i]