MinT - Best Known (253−127, 253, s)-Nets in Base 4

Best Known (253−127, 253, s)-Nets in Base 4

(253−127, 253, 130)-Net over F₄ — Constructive and digital

Digital (126, 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]

(253−127, 253, 176)-Net over F₄ — Digital

Digital (126, 253, 176)-net over F₄, using

t-expansion [i] based on digital (125, 253, 176)-net over F₄, using
- net from sequence [i] based on digital (125, 175)-sequence over F₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 125 and N(F) ≥ 176, using
    - Drinfeld modules of rank 1 [i]

(253−127, 253, 2022)-Net in Base 4 — Upper bound on s

There is no (126, 253, 2023)-net in base 4, because

1 times m-reduction [i] would yield (126, 252, 2023)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 53 341072 093706 197607 589057 040079 553383 665182 143993 979952 389666 793891 668292 573381 509269 921498 501346 072721 967360 983978 773598 731930 094302 051176 939161 472800 > 4²⁵² [i]