MinT - Best Known (192, 192+45, s)-Nets in Base 4

Best Known (192, 192+45, s)-Nets in Base 4

(192, 192+45, 1539)-Net over F₄ — Constructive and digital

Digital (192, 237, 1539)-net over F₄, using

9 times m-reduction [i] based on digital (192, 246, 1539)-net over F₄, using
- trace code for nets [i] based on digital (28, 82, 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]

(192, 192+45, 11307)-Net over F₄ — Digital

Digital (192, 237, 11307)-net over F₄, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(4²³⁷, 11307, F₄, 45) (dual of [11307, 11070, 46]-code), using
- discarding factors / shortening the dual code based on linear OA(4²³⁷, 16410, F₄, 45) (dual of [16410, 16173, 46]-code), using
  - constructionÂ X applied to C_e(44) âŠ‚ C_e(40) [i] based on
    1. linear OA(4²³², 16384, F₄, 45) (dual of [16384, 16152, 46]-code), using an extension C_e(44) of the primitive narrow-sense BCH-code C(I) with length 16383Â = 4⁷âˆ’1, defining interval IÂ =Â [1,44], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 45 [i]
    2. linear OA(4²¹¹, 16384, F₄, 41) (dual of [16384, 16173, 42]-code), using an extension C_e(40) of the primitive narrow-sense BCH-code C(I) with length 16383Â = 4⁷âˆ’1, defining interval IÂ =Â [1,40], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 41 [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]

(192, 192+45, large)-Net in Base 4 — Upper bound on s

There is no (192, 237, large)-net in base 4, because

43 times m-reduction [i] would yield (192, 194, large)-net in base 4, but
- mutually orthogonal hypercube bound [i]