MinT - Best Known (253−141, 253, s)-Nets in Base 4

Best Known (253−141, 253, s)-Nets in Base 4

(253−141, 253, 130)-Net over F₄ — Constructive and digital

Digital (112, 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]

(253−141, 253, 165)-Net over F₄ — Digital

Digital (112, 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]

(253−141, 253, 1261)-Net in Base 4 — Upper bound on s

There is no (112, 253, 1262)-net in base 4, because

1 times m-reduction [i] would yield (112, 252, 1262)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 54 277081 512930 460973 495768 682526 002683 254512 030136 824616 866659 788984 809971 683228 958037 940760 456853 275217 014850 793971 534887 464858 630472 181441 045424 276876 > 4²⁵² [i]