MinT - Best Known (221−109, 221, s)-Nets in Base 4

Best Known (221−109, 221, s)-Nets in Base 4

(221−109, 221, 130)-Net over F₄ — Constructive and digital

Digital (112, 221, 130)-net over F₄, using

t-expansion [i] based on digital (105, 221, 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]

(221−109, 221, 165)-Net over F₄ — Digital

Digital (112, 221, 165)-net over F₄, using

t-expansion [i] based on digital (109, 221, 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]

(221−109, 221, 1938)-Net in Base 4 — Upper bound on s

There is no (112, 221, 1939)-net in base 4, because

1 times m-reduction [i] would yield (112, 220, 1939)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 2 871484 188720 914523 412112 988792 550392 661070 473010 984694 549325 918149 468731 136160 076715 526813 598210 603343 863712 309012 078684 535922 136352 > 4²²⁰ [i]