MinT - Best Known (162, 162+35, s)-Nets in Base 4

Best Known (162, 162+35, s)-Nets in Base 4

(162, 162+35, 1539)-Net over F₄ — Constructive and digital

Digital (162, 197, 1539)-net over F₄, using

4 times m-reduction [i] based on digital (162, 201, 1539)-net over F₄, using
- trace code for nets [i] based on digital (28, 67, 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]

(162, 162+35, 16441)-Net over F₄ — Digital

Digital (162, 197, 16441)-net over F₄, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(4¹⁹⁷, 16441, F₄, 35) (dual of [16441, 16244, 36]-code), using
- constructionÂ X applied to C([0,17]) âŠ‚ C([0,13]) [i] based on
  1. linear OA(4¹⁸³, 16385, F₄, 35) (dual of [16385, 16202, 36]-code), using the expurgated narrow-sense BCH-code C(I) with length 16385Â | 4¹⁴âˆ’1, defining interval IÂ =Â [0,17], and minimum distance dÂ â‰¥ |{−17,−16,…,17}|+1Â =Â 36 (BCH-bound) [i]
  2. linear OA(4¹⁴¹, 16385, F₄, 27) (dual of [16385, 16244, 28]-code), using the expurgated narrow-sense BCH-code C(I) with length 16385Â | 4¹⁴âˆ’1, defining interval IÂ =Â [0,13], and minimum distance dÂ â‰¥ |{−13,−12,…,13}|+1Â =Â 28 (BCH-bound) [i]
  3. linear OA(4¹⁴, 56, F₄, 7) (dual of [56, 42, 8]-code), using
    - discarding factors / shortening the dual code based on linear OA(4¹⁴, 65, F₄, 7) (dual of [65, 51, 8]-code), using
      - a â€œGraXXâ€ code from Grasslâ€™s database [i]

(162, 162+35, large)-Net in Base 4 — Upper bound on s

There is no (162, 197, large)-net in base 4, because

33 times m-reduction [i] would yield (162, 164, large)-net in base 4, but
- mutually orthogonal hypercube bound [i]