MinT - Best Known (236−43, 236, s)-Nets in Base 4

Best Known (236−43, 236, s)-Nets in Base 4

(236−43, 236, 1549)-Net over F₄ — Constructive and digital

Digital (193, 236, 1549)-net over F₄, using

(u,Â u+v)-construction [i] based on
1. digital (2, 23, 10)-net over F₄, using
  - net from sequence [i] based on digital (2, 9)-sequence over F₄, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 2 and N(F) ≥ 10, using
      - an explicitly constructive algebraic function field [i]
2. digital (170, 213, 1539)-net over F₄, using
  - trace code for nets [i] based on digital (28, 71, 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]

(236−43, 236, 15159)-Net over F₄ — Digital

Digital (193, 236, 15159)-net over F₄, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(4²³⁶, 15159, F₄, 43) (dual of [15159, 14923, 44]-code), using
- discarding factors / shortening the dual code based on linear OA(4²³⁶, 16430, F₄, 43) (dual of [16430, 16194, 44]-code), using
  - 1 times code embedding in larger space [i] based on linear OA(4²³⁵, 16429, F₄, 43) (dual of [16429, 16194, 44]-code), using
    - constructionÂ X applied to C_e(42) âŠ‚ C_e(36) [i] based on
      1. linear OA(4²²⁵, 16384, F₄, 43) (dual of [16384, 16159, 44]-code), using an extension C_e(42) of the primitive narrow-sense BCH-code C(I) with length 16383Â = 4⁷âˆ’1, defining interval IÂ =Â [1,42], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 43 [i]
      2. 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]
      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]

(236−43, 236, large)-Net in Base 4 — Upper bound on s

There is no (193, 236, large)-net in base 4, because

41 times m-reduction [i] would yield (193, 195, large)-net in base 4, but
- mutually orthogonal hypercube bound [i]