MinT - Best Known (52, 52+167, s)-Nets in Base 4

Best Known (52, 52+167, s)-Nets in Base 4

(52, 52+167, 66)-Net over F₄ — Constructive and digital

Digital (52, 219, 66)-net over F₄, using

t-expansion [i] based on digital (49, 219, 66)-net over F₄, using
- net from sequence [i] based on digital (49, 65)-sequence over F₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 49 and N(F) ≥ 66, using
    - T₆ from the second tower of function fields by GarcÃa and Stichtenoth over F₄ [i]

(52, 52+167, 91)-Net over F₄ — Digital

Digital (52, 219, 91)-net over F₄, using

t-expansion [i] based on digital (50, 219, 91)-net over F₄, using
- net from sequence [i] based on digital (50, 90)-sequence over F₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 50 and N(F) ≥ 91, using
    - a curve from the manYPoints database [i]

(52, 52+167, 216)-Net over F₄ — Upper bound on s (digital)

There is no digital (52, 219, 217)-net over F₄, because

7 times m-reduction [i] would yield digital (52, 212, 217)-net over F₄, but
- extracting embedded orthogonal array [i] would yield linear OA(4²¹², 217, F₄, 160) (dual of [217, 5, 161]-code), but
  - residual code [i] would yield linear OA(4⁵², 56, F₄, 40) (dual of [56, 4, 41]-code), but
    - a result by Hill and Landjev [i]
    - improvement by Maruta on the Griesmer bound [i]

(52, 52+167, 222)-Net in Base 4 — Upper bound on s

There is no (52, 219, 223)-net in base 4, because

extracting embedded orthogonal array [i] would yield OA(4²¹⁹, 223, S₄, 167), but
- the (dual) Plotkin bound shows that MÂ â‰¥ 5 678427 533559 428832 416592 249125 035424 637823 130369 672345 949142 181098 744438 385921 275985 867583 701277 855943 457200 048954 515105 739075 223552 / 7 > 4²¹⁹ [i]