MinT - Best Known (188−33, 188, s)-Nets in Base 4

Best Known (188−33, 188, s)-Nets in Base 4

(188−33, 188, 1539)-Net over F₄ — Constructive and digital

Digital (155, 188, 1539)-net over F₄, using

t-expansion [i] based on 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]

(188−33, 188, 16450)-Net over F₄ — Digital

Digital (155, 188, 16450)-net over F₄, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(4¹⁸⁸, 16450, F₄, 33) (dual of [16450, 16262, 34]-code), using
- constructionÂ X applied to C_e(32) âŠ‚ C_e(22) [i] based on
  1. linear OA(4¹⁶⁹, 16384, F₄, 33) (dual of [16384, 16215, 34]-code), using an extension C_e(32) of the primitive narrow-sense BCH-code C(I) with length 16383Â = 4⁷âˆ’1, defining interval IÂ =Â [1,32], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 33 [i]
  2. linear OA(4¹²⁰, 16384, F₄, 23) (dual of [16384, 16264, 24]-code), using an extension C_e(22) of the primitive narrow-sense BCH-code C(I) with length 16383Â = 4⁷âˆ’1, defining interval IÂ =Â [1,22], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 23 [i]
  3. linear OA(4¹⁹, 66, F₄, 9) (dual of [66, 47, 10]-code), using
    - a code from Grasslâ€™s database [i]

(188−33, 188, large)-Net in Base 4 — Upper bound on s

There is no (155, 188, large)-net in base 4, because

31 times m-reduction [i] would yield (155, 157, large)-net in base 4, but
- mutually orthogonal hypercube bound [i]