MinT - Best Known (253−142, 253, s)-Nets in Base 4

Best Known (253−142, 253, s)-Nets in Base 4

(253−142, 253, 130)-Net over F₄ — Constructive and digital

Digital (111, 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]

(253−142, 253, 165)-Net over F₄ — Digital

Digital (111, 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]

(253−142, 253, 1212)-Net in Base 4 — Upper bound on s

There is no (111, 253, 1213)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 214 765707 253620 948656 889475 352180 294042 248221 027226 555840 422315 022202 944930 451897 076002 128530 447230 105500 112056 912338 263854 681626 474171 409805 675592 523600 > 4²⁵³ [i]