MinT - Best Known (191−36, 191, s)-Nets in Base 4

Best Known (191−36, 191, s)-Nets in Base 4

(191−36, 191, 1126)-Net over F₄ — Constructive and digital

Digital (155, 191, 1126)-net over F₄, using

4¹ times duplication [i] based on digital (154, 190, 1126)-net over F₄, using
- (u,Â u+v)-construction [i] based on
  1. digital (28, 46, 98)-net over F₄, using
    - trace code for nets [i] based on digital (5, 23, 49)-net over F₁₆, using
      - net from sequence [i] based on digital (5, 48)-sequence over F₁₆, using
        Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₆ with g(F) = 5 and N(F) ≥ 49, using
        a function field by SÃ©mirat [i]
  2. digital (108, 144, 1028)-net over F₄, using
    - trace code for nets [i] based on digital (0, 36, 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]

(191−36, 191, 10415)-Net over F₄ — Digital

Digital (155, 191, 10415)-net over F₄, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(4¹⁹¹, 10415, F₄, 36) (dual of [10415, 10224, 37]-code), using
- discarding factors / shortening the dual code based on linear OA(4¹⁹¹, 16399, F₄, 36) (dual of [16399, 16208, 37]-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(4¹, 15, F₄, 1) (dual of [15, 14, 2]-code), using
      - discarding factors / shortening the dual code based on linear OA(4¹, s, F₄, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
        parity-check code [i]

(191−36, 191, 6159326)-Net in Base 4 — Upper bound on s

There is no (155, 191, 6159327)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 9 850509 055809 658863 746294 398096 837947 733556 181530 368177 830843 919399 895434 990494 350718 895143 618646 222434 489132 981877 > 4¹⁹¹ [i]