MinT - Best Known (234−121, 234, s)-Nets in Base 4

Best Known (234−121, 234, s)-Nets in Base 4

(234−121, 234, 130)-Net over F₄ — Constructive and digital

Digital (113, 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]

(234−121, 234, 165)-Net over F₄ — Digital

Digital (113, 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]

(234−121, 234, 1634)-Net in Base 4 — Upper bound on s

There is no (113, 234, 1635)-net in base 4, because

1 times m-reduction [i] would yield (113, 233, 1635)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 193 593356 420225 499552 070465 337455 556724 180378 738249 249772 862574 064224 729135 161177 955570 303369 976307 376376 757175 979313 432629 750767 562779 443440 > 4²³³ [i]