MinT - Best Known (152, 185, s)-Nets in Base 4

Best Known (152, 185, s)-Nets in Base 4

(152, 185, 1539)-Net over F₄ — Constructive and digital

Digital (152, 185, 1539)-net over F₄, using

1 times m-reduction [i] based on digital (152, 186, 1539)-net over F₄, using
- trace code for nets [i] based on digital (28, 62, 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]

(152, 185, 15479)-Net over F₄ — Digital

Digital (152, 185, 15479)-net over F₄, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(4¹⁸⁵, 15479, F₄, 33) (dual of [15479, 15294, 34]-code), using
- discarding factors / shortening the dual code based on linear OA(4¹⁸⁵, 16402, F₄, 33) (dual of [16402, 16217, 34]-code), using
  - constructionÂ X applied to C([0,16]) âŠ‚ C([1,16]) [i] based on
    1. 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]
    2. linear OA(4¹⁶⁸, 16385, F₄, 16) (dual of [16385, 16217, 17]-code), using the narrow-sense BCH-code C(I) with length 16385Â | 4¹⁴âˆ’1, defining interval IÂ =Â [1,16], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 17 [i]
    3. linear OA(4¹⁶, 17, F₄, 16) (dual of [17, 1, 17]-code or 17-arc in PG(15,4)), using
      - dual of repetition code with length 17 [i]

(152, 185, large)-Net in Base 4 — Upper bound on s

There is no (152, 185, large)-net in base 4, because

31 times m-reduction [i] would yield (152, 154, large)-net in base 4, but
- mutually orthogonal hypercube bound [i]