MinT - Best Known (164, 164+36, s)-Nets in Base 4

Best Known (164, 164+36, s)-Nets in Base 4

(164, 164+36, 1539)-Net over F₄ — Constructive and digital

Digital (164, 200, 1539)-net over F₄, using

4 times m-reduction [i] based on 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]

(164, 164+36, 15045)-Net over F₄ — Digital

Digital (164, 200, 15045)-net over F₄, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(4²⁰⁰, 15045, F₄, 36) (dual of [15045, 14845, 37]-code), using
- discarding factors / shortening the dual code based on linear OA(4²⁰⁰, 16406, F₄, 36) (dual of [16406, 16206, 37]-code), using
  - constructionÂ X applied to C([0,18]) âŠ‚ C([0,16]) [i] based on
    1. linear OA(4¹⁹⁷, 16385, F₄, 37) (dual of [16385, 16188, 38]-code), using the expurgated narrow-sense BCH-code C(I) with length 16385Â | 4¹⁴âˆ’1, defining interval IÂ =Â [0,18], and minimum distance dÂ â‰¥ |{−18,−17,…,18}|+1Â =Â 38 (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(4³, 21, F₄, 2) (dual of [21, 18, 3]-code), using
      - Hamming code H(3,4) [i]

(164, 164+36, large)-Net in Base 4 — Upper bound on s

There is no (164, 200, large)-net in base 4, because

34 times m-reduction [i] would yield (164, 166, large)-net in base 4, but
- mutually orthogonal hypercube bound [i]