MinT - Best Known (250−127, 250, s)-Nets in Base 4

Best Known (250−127, 250, s)-Nets in Base 4

(250−127, 250, 130)-Net over F₄ — Constructive and digital

Digital (123, 250, 130)-net over F₄, using

t-expansion [i] based on digital (105, 250, 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]

(250−127, 250, 168)-Net over F₄ — Digital

Digital (123, 250, 168)-net over F₄, using

t-expansion [i] based on digital (115, 250, 168)-net over F₄, using
- net from sequence [i] based on digital (115, 167)-sequence over F₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 115 and N(F) ≥ 168, using
    - Drinfeld modules of rank 1 [i]

(250−127, 250, 1889)-Net in Base 4 — Upper bound on s

There is no (123, 250, 1890)-net in base 4, because

1 times m-reduction [i] would yield (123, 249, 1890)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 819805 540007 729851 637392 037853 703979 896217 203971 777375 036444 149320 891211 318859 323838 707100 227028 615170 114900 175316 090818 597333 138094 338697 521133 160608 > 4²⁴⁹ [i]