MinT - Best Known (252−139, 252, s)-Nets in Base 4

Best Known (252−139, 252, s)-Nets in Base 4

(252−139, 252, 130)-Net over F₄ — Constructive and digital

Digital (113, 252, 130)-net over F₄, using

t-expansion [i] based on digital (105, 252, 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]

(252−139, 252, 165)-Net over F₄ — Digital

Digital (113, 252, 165)-net over F₄, using

t-expansion [i] based on digital (109, 252, 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]

(252−139, 252, 1313)-Net in Base 4 — Upper bound on s

There is no (113, 252, 1314)-net in base 4, because

1 times m-reduction [i] would yield (113, 251, 1314)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 13 312960 665765 119382 020978 689590 325859 815896 389779 835398 223733 805924 123933 230385 955458 431781 797930 798431 985272 998430 602319 318429 463199 552987 391027 459680 > 4²⁵¹ [i]