MinT - Best Known (253−139, 253, s)-Nets in Base 4

Best Known (253−139, 253, s)-Nets in Base 4

(253−139, 253, 130)-Net over F₄ — Constructive and digital

Digital (114, 253, 130)-net over F₄, using

t-expansion [i] based on digital (105, 253, 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]

(253−139, 253, 165)-Net over F₄ — Digital

Digital (114, 253, 165)-net over F₄, using

t-expansion [i] based on digital (109, 253, 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]

(253−139, 253, 1341)-Net in Base 4 — Upper bound on s

There is no (114, 253, 1342)-net in base 4, because

1 times m-reduction [i] would yield (114, 252, 1342)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 53 811214 406731 602796 944444 790021 963848 141111 337645 481406 357843 115300 301151 148138 969745 037740 174639 313857 123189 338934 774808 780593 971443 883400 989758 087460 > 4²⁵² [i]