MinT - Best Known (193−37, 193, s)-Nets in Base 4

Best Known (193−37, 193, s)-Nets in Base 4

(193−37, 193, 1118)-Net over F₄ — Constructive and digital

Digital (156, 193, 1118)-net over F₄, using

4¹ times duplication [i] based on digital (155, 192, 1118)-net over F₄, using
- (u,Â u+v)-construction [i] based on
  1. digital (26, 44, 90)-net over F₄, using
    - trace code for nets [i] based on digital (4, 22, 45)-net over F₁₆, using
      - net from sequence [i] based on digital (4, 44)-sequence over F₁₆, using
        Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₆ with g(F) = 4 and N(F) ≥ 45, using
        a function field by GarcÃa and Quoos [i]
  2. digital (111, 148, 1028)-net over F₄, using
    - trace code for nets [i] based on digital (0, 37, 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]

(193−37, 193, 9280)-Net over F₄ — Digital

Digital (156, 193, 9280)-net over F₄, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(4¹⁹³, 9280, F₄, 37) (dual of [9280, 9087, 38]-code), using
- discarding factors / shortening the dual code based on linear OA(4¹⁹³, 16401, F₄, 37) (dual of [16401, 16208, 38]-code), using
  - constructionÂ X applied to C_e(36) âŠ‚ C_e(33) [i] based on
    1. linear OA(4¹⁹⁰, 16384, F₄, 37) (dual of [16384, 16194, 38]-code), using an extension C_e(36) of the primitive narrow-sense BCH-code C(I) with length 16383Â = 4⁷âˆ’1, defining interval IÂ =Â [1,36], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 37 [i]
    2. linear OA(4¹⁷⁶, 16384, F₄, 34) (dual of [16384, 16208, 35]-code), using an extension C_e(33) of the primitive narrow-sense BCH-code C(I) with length 16383Â = 4⁷âˆ’1, defining interval IÂ =Â [1,33], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 34 [i]
    3. linear OA(4³, 17, F₄, 2) (dual of [17, 14, 3]-code), using
      - discarding factors / shortening the dual code based on linear OA(4³, 21, F₄, 2) (dual of [21, 18, 3]-code), using
        Hamming code H(3,4) [i]

(193−37, 193, 6652441)-Net in Base 4 — Upper bound on s

There is no (156, 193, 6652442)-net in base 4, because

1 times m-reduction [i] would yield (156, 192, 6652442)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 39 402007 199368 546937 129846 078118 128257 299325 694491 085317 496968 886664 911236 755980 013343 042084 292188 257518 443028 510836 > 4¹⁹² [i]