MinT - Best Known (199, 199+44, s)-Nets in Base 4

Best Known (199, 199+44, s)-Nets in Base 4

(199, 199+44, 1556)-Net over F₄ — Constructive and digital

Digital (199, 243, 1556)-net over F₄, using

(u,Â u+v)-construction [i] based on
1. digital (5, 27, 17)-net over F₄, using
  - net from sequence [i] based on digital (5, 16)-sequence over F₄, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 5 and N(F) ≥ 17, using
      - an explicitly constructive algebraic function field [i]
2. digital (172, 216, 1539)-net over F₄, using
  - trace code for nets [i] based on digital (28, 72, 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]

(199, 199+44, 16173)-Net over F₄ — Digital

Digital (199, 243, 16173)-net over F₄, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(4²⁴³, 16173, F₄, 44) (dual of [16173, 15930, 45]-code), using
- discarding factors / shortening the dual code based on linear OA(4²⁴³, 16430, F₄, 44) (dual of [16430, 16187, 45]-code), using
  - 1 times code embedding in larger space [i] based on linear OA(4²⁴², 16429, F₄, 44) (dual of [16429, 16187, 45]-code), using
    - constructionÂ X applied to C_e(44) âŠ‚ C_e(37) [i] based on
      1. linear OA(4²³², 16384, F₄, 45) (dual of [16384, 16152, 46]-code), using an extension C_e(44) of the primitive narrow-sense BCH-code C(I) with length 16383Â = 4⁷âˆ’1, defining interval IÂ =Â [1,44], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 45 [i]
      2. 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]
      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]

(199, 199+44, large)-Net in Base 4 — Upper bound on s

There is no (199, 243, large)-net in base 4, because

42 times m-reduction [i] would yield (199, 201, large)-net in base 4, but
- mutually orthogonal hypercube bound [i]