MinT - Best Known (257−135, 257, s)-Nets in Base 4

Best Known (257−135, 257, s)-Nets in Base 4

(257−135, 257, 130)-Net over F₄ — Constructive and digital

Digital (122, 257, 130)-net over F₄, using

t-expansion [i] based on digital (105, 257, 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]

(257−135, 257, 168)-Net over F₄ — Digital

Digital (122, 257, 168)-net over F₄, using

t-expansion [i] based on digital (115, 257, 168)-net over F₄, using
- net from sequence [i] based on digital (115, 167)-sequence over F₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 115 and N(F) ≥ 168, using
    - Drinfeld modules of rank 1 [i]

(257−135, 257, 1661)-Net in Base 4 — Upper bound on s

There is no (122, 257, 1662)-net in base 4, because

1 times m-reduction [i] would yield (122, 256, 1662)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 13461 214436 275878 371576 095972 617358 845253 607030 330028 282900 380410 986929 446387 771547 500329 239097 353176 085340 559983 975762 363780 713187 305146 769223 090388 389550 > 4²⁵⁶ [i]