MinT - Best Known (166, 166+39, s)-Nets in Base 4

Best Known (166, 166+39, s)-Nets in Base 4

(166, 166+39, 1539)-Net over F₄ — Constructive and digital

Digital (166, 205, 1539)-net over F₄, using

2 times m-reduction [i] based on digital (166, 207, 1539)-net over F₄, using
- trace code for nets [i] based on digital (28, 69, 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]

(166, 166+39, 10162)-Net over F₄ — Digital

Digital (166, 205, 10162)-net over F₄, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(4²⁰⁵, 10162, F₄, 39) (dual of [10162, 9957, 40]-code), using
- discarding factors / shortening the dual code based on linear OA(4²⁰⁵, 16399, F₄, 39) (dual of [16399, 16194, 40]-code), using
  - constructionÂ X applied to C_e(38) âŠ‚ C_e(36) [i] based on
    1. linear OA(4²⁰⁴, 16384, F₄, 39) (dual of [16384, 16180, 40]-code), using an extension C_e(38) of the primitive narrow-sense BCH-code C(I) with length 16383Â = 4⁷âˆ’1, defining interval IÂ =Â [1,38], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 39 [i]
    2. linear OA(4¹⁹⁰, 16384, F₄, 37) (dual of [16384, 16194, 38]-code), using an extension C_e(36) of the primitive narrow-sense BCH-code C(I) with length 16383Â = 4⁷âˆ’1, defining interval IÂ =Â [1,36], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 37 [i]
    3. linear OA(4¹, 15, F₄, 1) (dual of [15, 14, 2]-code), using
      - discarding factors / shortening the dual code based on linear OA(4¹, s, F₄, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
        parity-check code [i]

(166, 166+39, 7696910)-Net in Base 4 — Upper bound on s

There is no (166, 205, 7696911)-net in base 4, because

1 times m-reduction [i] would yield (166, 204, 7696911)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 661 056360 273907 468922 020057 496179 274620 361085 409425 788875 602807 585887 594125 008891 403877 064407 077090 765297 513191 232709 943040 > 4²⁰⁴ [i]