MinT - Best Known (209, 209+47, s)-Nets in Base 4

Best Known (209, 209+47, s)-Nets in Base 4

(209, 209+47, 1560)-Net over F₄ — Constructive and digital

Digital (209, 256, 1560)-net over F₄, using

4¹ times duplication [i] based on digital (208, 255, 1560)-net over F₄, using
- (u,Â u+v)-construction [i] based on
  1. digital (7, 30, 21)-net over F₄, using
    - net from sequence [i] based on digital (7, 20)-sequence over F₄, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 7 and N(F) ≥ 21, using
        an explicitly constructive algebraic function field [i]
  2. digital (178, 225, 1539)-net over F₄, using
    - trace code for nets [i] based on digital (28, 75, 513)-net over F₆₄, using
      - net from sequence [i] based on digital (28, 512)-sequence over F₆₄, using
        Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 28 and N(F) ≥ 513, using
        the Hermitian function field over F₆₄ [i]

(209, 209+47, 15125)-Net over F₄ — Digital

Digital (209, 256, 15125)-net over F₄, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(4²⁵⁶, 15125, F₄, 47) (dual of [15125, 14869, 48]-code), using
- discarding factors / shortening the dual code based on linear OA(4²⁵⁶, 16429, F₄, 47) (dual of [16429, 16173, 48]-code), using
  - constructionÂ X applied to C_e(46) âŠ‚ C_e(40) [i] based on
    1. linear OA(4²⁴⁶, 16384, F₄, 47) (dual of [16384, 16138, 48]-code), using an extension C_e(46) of the primitive narrow-sense BCH-code C(I) with length 16383Â = 4⁷âˆ’1, defining interval IÂ =Â [1,46], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 47 [i]
    2. linear OA(4²¹¹, 16384, F₄, 41) (dual of [16384, 16173, 42]-code), using an extension C_e(40) of the primitive narrow-sense BCH-code C(I) with length 16383Â = 4⁷âˆ’1, defining interval IÂ =Â [1,40], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 41 [i]
    3. linear OA(4¹⁰, 45, F₄, 5) (dual of [45, 35, 6]-code), using
      - discarding factors / shortening the dual code based on linear OA(4¹⁰, 63, F₄, 5) (dual of [63, 53, 6]-code), using
        the primitive expurgated narrow-sense BCH-code C(I) with length 63Â = 4³âˆ’1, defining interval IÂ =Â [0,3], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 6 [i]

(209, 209+47, large)-Net in Base 4 — Upper bound on s

There is no (209, 256, large)-net in base 4, because

45 times m-reduction [i] would yield (209, 211, large)-net in base 4, but
- mutually orthogonal hypercube bound [i]