MinT - Best Known (166, 166+38, s)-Nets in Base 4

Best Known (166, 166+38, s)-Nets in Base 4

(166, 166+38, 1539)-Net over F₄ — Constructive and digital

Digital (166, 204, 1539)-net over F₄, using

3 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+38, 11790)-Net over F₄ — Digital

Digital (166, 204, 11790)-net over F₄, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(4²⁰⁴, 11790, F₄, 38) (dual of [11790, 11586, 39]-code), using
- discarding factors / shortening the dual code based on linear OA(4²⁰⁴, 16419, F₄, 38) (dual of [16419, 16215, 39]-code), using
  - constructionÂ X applied to C_e(37) âŠ‚ C_e(32) [i] based on
    1. linear OA(4¹⁹⁷, 16384, F₄, 38) (dual of [16384, 16187, 39]-code), using an extension C_e(37) of the primitive narrow-sense BCH-code C(I) with length 16383Â = 4⁷âˆ’1, defining interval IÂ =Â [1,37], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 38 [i]
    2. 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]
    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]

(166, 166+38, 7696910)-Net in Base 4 — Upper bound on s

There is no (166, 204, 7696911)-net in base 4, because

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]