MinT - Best Known (165, 165+35, s)-Nets in Base 4

Best Known (165, 165+35, s)-Nets in Base 4

(165, 165+35, 1539)-Net over F₄ — Constructive and digital

Digital (165, 200, 1539)-net over F₄, using

t-expansion [i] based on digital (164, 200, 1539)-net over F₄, using
- 4 times m-reduction [i] based on digital (164, 204, 1539)-net over F₄, using
  - trace code for nets [i] based on digital (28, 68, 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]

(165, 165+35, 16447)-Net over F₄ — Digital

Digital (165, 200, 16447)-net over F₄, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(4²⁰⁰, 16447, F₄, 35) (dual of [16447, 16247, 36]-code), using
- constructionÂ X applied to C_e(34) âŠ‚ C_e(25) [i] based on
  1. linear OA(4¹⁸³, 16384, F₄, 35) (dual of [16384, 16201, 36]-code), using an extension C_e(34) of the primitive narrow-sense BCH-code C(I) with length 16383Â = 4⁷âˆ’1, defining interval IÂ =Â [1,34], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 35 [i]
  2. linear OA(4¹³⁴, 16384, F₄, 26) (dual of [16384, 16250, 27]-code), using an extension C_e(25) of the primitive narrow-sense BCH-code C(I) with length 16383Â = 4⁷âˆ’1, defining interval IÂ =Â [1,25], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 26 [i]
  3. linear OA(4¹⁷, 63, F₄, 8) (dual of [63, 46, 9]-code), using
    - a code from Grasslâ€™s database [i]

(165, 165+35, large)-Net in Base 4 — Upper bound on s

There is no (165, 200, large)-net in base 4, because

33 times m-reduction [i] would yield (165, 167, large)-net in base 4, but
- mutually orthogonal hypercube bound [i]