MinT - Best Known (197−38, 197, s)-Nets in Base 4

Best Known (197−38, 197, s)-Nets in Base 4

(197−38, 197, 1104)-Net over F₄ — Constructive and digital

Digital (159, 197, 1104)-net over F₄, using

4¹ times duplication [i] based on digital (158, 196, 1104)-net over F₄, using
- (u,Â u+v)-construction [i] based on
  1. digital (25, 44, 76)-net over F₄, using
    - trace code for nets [i] based on digital (3, 22, 38)-net over F₁₆, using
      - net from sequence [i] based on digital (3, 37)-sequence over F₁₆, using
        Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₆ with g(F) = 3 and N(F) ≥ 38, using
        a function field by SÃ©mirat [i]
  2. digital (114, 152, 1028)-net over F₄, using
    - trace code for nets [i] based on digital (0, 38, 257)-net over F₂₅₆, using
      - net from sequence [i] based on digital (0, 256)-sequence over F₂₅₆, using
        generalized Faure sequence [i]
        Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₅₆ with g(F) = 0 and N(F) ≥ 257, using
        the rational function field F₂₅₆(x) [i]
        Niederreiter sequence [i]

(197−38, 197, 8997)-Net over F₄ — Digital

Digital (159, 197, 8997)-net over F₄, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(4¹⁹⁷, 8997, F₄, 38) (dual of [8997, 8800, 39]-code), using
- discarding factors / shortening the dual code based on linear OA(4¹⁹⁷, 16384, F₄, 38) (dual of [16384, 16187, 39]-code), using
  - an extension C_e(37) of the primitive narrow-sense BCH-code C(I) with length 16383Â = 4⁷âˆ’1, defining interval IÂ =Â [1,37], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 38 [i]

(197−38, 197, 4618535)-Net in Base 4 — Upper bound on s

There is no (159, 197, 4618536)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 40347 714889 982673 480351 492954 905104 997225 517358 389872 638652 955154 490528 062291 233578 737143 935114 097042 334102 147865 904790 > 4¹⁹⁷ [i]