MinT - Best Known (157, 157+35, s)-Nets in Base 4

Best Known (157, 157+35, s)-Nets in Base 4

(157, 157+35, 1539)-Net over F₄ — Constructive and digital

Digital (157, 192, 1539)-net over F₄, using

t-expansion [i] based on digital (156, 192, 1539)-net over F₄, using
- trace code for nets [i] based on digital (28, 64, 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]

(157, 157+35, 13367)-Net over F₄ — Digital

Digital (157, 192, 13367)-net over F₄, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(4¹⁹², 13367, F₄, 35) (dual of [13367, 13175, 36]-code), using
- discarding factors / shortening the dual code based on linear OA(4¹⁹², 16409, F₄, 35) (dual of [16409, 16217, 36]-code), using
  - constructionÂ X applied to C([0,17]) âŠ‚ C([0,14]) [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(4¹⁵⁵, 16385, F₄, 29) (dual of [16385, 16230, 30]-code), using the expurgated narrow-sense BCH-code C(I) with length 16385Â | 4¹⁴âˆ’1, defining interval IÂ =Â [0,14], and minimum distance dÂ â‰¥ |{−14,−13,…,14}|+1Â =Â 30 (BCH-bound) [i]
    3. linear OA(4⁹, 24, F₄, 5) (dual of [24, 15, 6]-code), using
      - discarding factors / shortening the dual code based on linear OA(4⁹, 36, F₄, 5) (dual of [36, 27, 6]-code), using
        codes constructed by inverting constructionÂ Y₁ [i]

(157, 157+35, large)-Net in Base 4 — Upper bound on s

There is no (157, 192, large)-net in base 4, because

33 times m-reduction [i] would yield (157, 159, large)-net in base 4, but
- mutually orthogonal hypercube bound [i]