MinT - Best Known (188−34, 188, s)-Nets in Base 4

Best Known (188−34, 188, s)-Nets in Base 4

(188−34, 188, 1539)-Net over F₄ — Constructive and digital

Digital (154, 188, 1539)-net over F₄, using

1 times m-reduction [i] based on digital (154, 189, 1539)-net over F₄, using
- trace code for nets [i] based on digital (28, 63, 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]

(188−34, 188, 14036)-Net over F₄ — Digital

Digital (154, 188, 14036)-net over F₄, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(4¹⁸⁸, 14036, F₄, 34) (dual of [14036, 13848, 35]-code), using
- discarding factors / shortening the dual code based on linear OA(4¹⁸⁸, 16424, F₄, 34) (dual of [16424, 16236, 35]-code), using
  - 5 times code embedding in larger space [i] based on linear OA(4¹⁸³, 16419, F₄, 34) (dual of [16419, 16236, 35]-code), using
    - constructionÂ X applied to C_e(33) âŠ‚ C_e(28) [i] based on
      1. linear OA(4¹⁷⁶, 16384, F₄, 34) (dual of [16384, 16208, 35]-code), using an extension C_e(33) of the primitive narrow-sense BCH-code C(I) with length 16383Â = 4⁷âˆ’1, defining interval IÂ =Â [1,33], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 34 [i]
      2. linear OA(4¹⁴⁸, 16384, F₄, 29) (dual of [16384, 16236, 30]-code), using an extension C_e(28) of the primitive narrow-sense BCH-code C(I) with length 16383Â = 4⁷âˆ’1, defining interval IÂ =Â [1,28], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 29 [i]
      3. linear OA(4⁷, 35, F₄, 4) (dual of [35, 28, 5]-code), using
        discarding factors / shortening the dual code based on linear OA(4⁷, 43, F₄, 4) (dual of [43, 36, 5]-code), using
        quaternary constacyclic codes with minimum distance 5 [i]

(188−34, 188, large)-Net in Base 4 — Upper bound on s

There is no (154, 188, large)-net in base 4, because

32 times m-reduction [i] would yield (154, 156, large)-net in base 4, but
- mutually orthogonal hypercube bound [i]