MinT - Best Known (225−40, 225, s)-Nets in Base 4

Best Known (225−40, 225, s)-Nets in Base 4

(225−40, 225, 1548)-Net over F₄ — Constructive and digital

Digital (185, 225, 1548)-net over F₄, using

(u,Â u+v)-construction [i] based on
1. digital (1, 21, 9)-net over F₄, using
  - net from sequence [i] based on digital (1, 8)-sequence over F₄, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 1 and N(F) ≥ 9, using
      - the Hermitian function field over F₄ [i]
2. digital (164, 204, 1539)-net over F₄, using
  - trace code for nets [i] based on digital (28, 68, 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]

(225−40, 225, 16442)-Net over F₄ — Digital

Digital (185, 225, 16442)-net over F₄, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(4²²⁵, 16442, F₄, 40) (dual of [16442, 16217, 41]-code), using
- constructionÂ X with VarÅ¡amov bound [i] based on
  1. linear OA(4²²⁴, 16440, F₄, 40) (dual of [16440, 16216, 41]-code), using
    - constructionÂ X applied to C([0,20]) âŠ‚ C([0,16]) [i] based on
      1. linear OA(4²¹¹, 16385, F₄, 41) (dual of [16385, 16174, 42]-code), using the expurgated narrow-sense BCH-code C(I) with length 16385Â | 4¹⁴âˆ’1, defining interval IÂ =Â [0,20], and minimum distance dÂ â‰¥ |{−20,−19,…,20}|+1Â =Â 42 (BCH-bound) [i]
      2. linear OA(4¹⁶⁹, 16385, F₄, 33) (dual of [16385, 16216, 34]-code), using the expurgated narrow-sense BCH-code C(I) with length 16385Â | 4¹⁴âˆ’1, defining interval IÂ =Â [0,16], and minimum distance dÂ â‰¥ |{−16,−15,…,16}|+1Â =Â 34 (BCH-bound) [i]
      3. linear OA(4¹³, 55, F₄, 6) (dual of [55, 42, 7]-code), using
        discarding factors / shortening the dual code based on linear OA(4¹³, 63, F₄, 6) (dual of [63, 50, 7]-code), using
        the primitive expurgated narrow-sense BCH-code C(I) with length 63Â = 4³âˆ’1, defining interval IÂ =Â [0,5], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 7 [i]
  2. linear OA(4²²⁴, 16441, F₄, 39) (dual of [16441, 16217, 40]-code), using Gilbertâ€“VarÅ¡amov bound and b^mÂ = 4²²⁴Â > V_b^sâˆ’1(kâˆ’1)Â = 39 637509 447789 024412 137436 951917 161835 935726 781549 691003 839799 715540 415922 004542 578898 489367 429927 129172 658550 707372 492455 369864 048061 [i]
  3. linear OA(4⁰, 1, F₄, 0) (dual of [1, 1, 1]-code), using
    - discarding factors / shortening the dual code based on linear OA(4⁰, s, F₄, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
      - OA with only one run / dual of code without redundancy [i]

(225−40, 225, large)-Net in Base 4 — Upper bound on s

There is no (185, 225, large)-net in base 4, because

38 times m-reduction [i] would yield (185, 187, large)-net in base 4, but
- mutually orthogonal hypercube bound [i]

Best Known (225−40, 225, s)-Nets in Base 4

(225−40, 225, 1548)-Net over F4 — Constructive and digital

(225−40, 225, 16442)-Net over F4 — Digital

(225−40, 225, large)-Net in Base 4 — Upper bound on s

(225−40, 225, 1548)-Net over F₄ — Constructive and digital

(225−40, 225, 16442)-Net over F₄ — Digital