MinT - Best Known (192−136, 192, s)-Nets in Base 4

Best Known (192−136, 192, s)-Nets in Base 4

(192−136, 192, 66)-Net over F₄ — Constructive and digital

Digital (56, 192, 66)-net over F₄, using

t-expansion [i] based on digital (49, 192, 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]

(192−136, 192, 91)-Net over F₄ — Digital

Digital (56, 192, 91)-net over F₄, using

t-expansion [i] based on digital (50, 192, 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]

(192−136, 192, 373)-Net over F₄ — Upper bound on s (digital)

There is no digital (56, 192, 374)-net over F₄, because

extracting embedded orthogonal array [i] would yield linear OA(4¹⁹², 374, F₄, 136) (dual of [374, 182, 137]-code), but
- residual code [i] would yield OA(4⁵⁶, 237, S₄, 34), but
  - the linear programming bound shows that MÂ â‰¥ 227 819482 170490 726334 286831 192651 607943 130369 749048 513507 843112 960000 000000 / 43777 691396 586787 052635 611770 906797 710989 > 4⁵⁶ [i]

(192−136, 192, 383)-Net in Base 4 — Upper bound on s

There is no (56, 192, 384)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 43 860403 798467 103649 180914 637083 396299 686470 018177 666335 089597 420813 317989 445493 990337 292757 637029 931056 430254 933431 > 4¹⁹² [i]