MinT - Best Known (164, 164+38, s)-Nets in Base 4

Best Known (164, 164+38, s)-Nets in Base 4

(164, 164+38, 1539)-Net over F₄ — Constructive and digital

Digital (164, 202, 1539)-net over F₄, using

2 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]

(164, 164+38, 10914)-Net over F₄ — Digital

Digital (164, 202, 10914)-net over F₄, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(4²⁰², 10914, F₄, 38) (dual of [10914, 10712, 39]-code), using
- discarding factors / shortening the dual code based on linear OA(4²⁰², 16410, F₄, 38) (dual of [16410, 16208, 39]-code), using
  - constructionÂ X applied to C_e(37) âŠ‚ C_e(33) [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(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]
    3. linear OA(4⁵, 26, F₄, 3) (dual of [26, 21, 4]-code or 26-cap in PG(4,4)), using
      - discarding factors / shortening the dual code based on linear OA(4⁵, 41, F₄, 3) (dual of [41, 36, 4]-code or 41-cap in PG(4,4)), using
        a cap found by Tallinii [i]

(164, 164+38, 6651836)-Net in Base 4 — Upper bound on s

There is no (164, 202, 6651837)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 41 316032 558542 605939 348683 680092 560499 405741 332910 478383 715370 274979 846770 843295 765530 270171 107300 081686 736040 627156 671740 > 4²⁰² [i]