MinT - Best Known (195−37, 195, s)-Nets in Base 4

Best Known (195−37, 195, s)-Nets in Base 4

(195−37, 195, 1539)-Net over F₄ — Constructive and digital

Digital (158, 195, 1539)-net over F₄, using

trace code for nets [i] based on digital (28, 65, 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]

(195−37, 195, 10048)-Net over F₄ — Digital

Digital (158, 195, 10048)-net over F₄, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(4¹⁹⁵, 10048, F₄, 37) (dual of [10048, 9853, 38]-code), using
- discarding factors / shortening the dual code based on linear OA(4¹⁹⁵, 16410, F₄, 37) (dual of [16410, 16215, 38]-code), using
  - constructionÂ X applied to C_e(36) âŠ‚ C_e(32) [i] based on
    1. 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]
    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(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]

(195−37, 195, 7760269)-Net in Base 4 — Upper bound on s

There is no (158, 195, 7760270)-net in base 4, because

1 times m-reduction [i] would yield (158, 194, 7760270)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 630 433297 066923 164757 579079 404852 373119 716312 447901 956681 635615 288403 985992 013172 042952 324122 060853 193875 531602 524112 > 4¹⁹⁴ [i]