MinT - Best Known (32−10, 32, s)-Nets in Base 4

Best Known (32−10, 32, s)-Nets in Base 4

(32−10, 32, 195)-Net over F₄ — Constructive and digital

Digital (22, 32, 195)-net over F₄, using

1 times m-reduction [i] based on digital (22, 33, 195)-net over F₄, using
- trace code for nets [i] based on digital (0, 11, 65)-net over F₆₄, using
  - net from sequence [i] based on digital (0, 64)-sequence over F₆₄, using
    - generalized Faure sequence [i]
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 0 and N(F) ≥ 65, using
      - the rational function field F₆₄(x) [i]
    - Niederreiter sequence [i]

(32−10, 32, 265)-Net over F₄ — Digital

Digital (22, 32, 265)-net over F₄, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(4³², 265, F₄, 10) (dual of [265, 233, 11]-code), using
- discarding factors / shortening the dual code based on linear OA(4³², 267, F₄, 10) (dual of [267, 235, 11]-code), using
  - constructionÂ X applied to C_e(9) âŠ‚ C_e(6) [i] based on
    1. linear OA(4²⁹, 256, F₄, 10) (dual of [256, 227, 11]-code), using an extension C_e(9) of the primitive narrow-sense BCH-code C(I) with length 255Â = 4⁴âˆ’1, defining interval IÂ =Â [1,9], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 10 [i]
    2. linear OA(4²¹, 256, F₄, 7) (dual of [256, 235, 8]-code), using an extension C_e(6) of the primitive narrow-sense BCH-code C(I) with length 255Â = 4⁴âˆ’1, defining interval IÂ =Â [1,6], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 7 [i]
    3. linear OA(4³, 11, F₄, 2) (dual of [11, 8, 3]-code), using
      - discarding factors / shortening the dual code based on linear OA(4³, 21, F₄, 2) (dual of [21, 18, 3]-code), using
        Hamming code H(3,4) [i]

(32−10, 32, 6189)-Net in Base 4 — Upper bound on s

There is no (22, 32, 6190)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 18 457116 882940 989385 > 4³² [i]