MinT - Best Known (168−127, 168, s)-Nets in Base 4

Best Known (168−127, 168, s)-Nets in Base 4

(168−127, 168, 56)-Net over F₄ — Constructive and digital

Digital (41, 168, 56)-net over F₄, using

t-expansion [i] based on digital (33, 168, 56)-net over F₄, using
- net from sequence [i] based on digital (33, 55)-sequence over F₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 33 and N(F) ≥ 56, using
    - F₅ from the tower of function fields by GarcÃa and Stichtenoth over F₄ [i]

(168−127, 168, 75)-Net over F₄ — Digital

Digital (41, 168, 75)-net over F₄, using

t-expansion [i] based on digital (40, 168, 75)-net over F₄, using
- net from sequence [i] based on digital (40, 74)-sequence over F₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 40 and N(F) ≥ 75, using
    - Drinfeld modules of rank 1 [i]

(168−127, 168, 175)-Net over F₄ — Upper bound on s (digital)

There is no digital (41, 168, 176)-net over F₄, because

3 times m-reduction [i] would yield digital (41, 165, 176)-net over F₄, but
- extracting embedded orthogonal array [i] would yield linear OA(4¹⁶⁵, 176, F₄, 124) (dual of [176, 11, 125]-code), but
  - residual code [i] would yield linear OA(4⁴¹, 51, F₄, 31) (dual of [51, 10, 32]-code), but
    - â€œGurâ€ bound on codes from Brouwerâ€™s database [i]

(168−127, 168, 270)-Net in Base 4 — Upper bound on s

There is no (41, 168, 271)-net in base 4, because

1 times m-reduction [i] would yield (41, 167, 271)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 38687 753687 950013 114019 933501 781639 551161 828910 856051 340875 204870 516290 917484 759483 142278 991458 140224 > 4¹⁶⁷ [i]