MinT - Best Known (27, 27+6, s)-Nets in Base 4

Best Known (27, 27+6, s)-Nets in Base 4

(27, 27+6, 21848)-Net over F₄ — Constructive and digital

Digital (27, 33, 21848)-net over F₄, using

net defined by OOA [i] based on linear OOA(4³³, 21848, F₄, 6, 6) (dual of [(21848, 6), 131055, 7]-NRT-code), using
- OA 3-folding and stacking [i] based on linear OA(4³³, 65544, F₄, 6) (dual of [65544, 65511, 7]-code), using
  - constructionÂ X applied to C_e(5) âŠ‚ C_e(4) [i] based on
    1. linear OA(4³³, 65536, F₄, 6) (dual of [65536, 65503, 7]-code), using an extension C_e(5) of the primitive narrow-sense BCH-code C(I) with length 65535Â = 4⁸âˆ’1, defining interval IÂ =Â [1,5], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 6 [i]
    2. linear OA(4²⁵, 65536, F₄, 5) (dual of [65536, 65511, 6]-code), using an extension C_e(4) of the primitive narrow-sense BCH-code C(I) with length 65535Â = 4⁸âˆ’1, defining interval IÂ =Â [1,4], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 5 [i]
    3. linear OA(4⁰, 8, F₄, 0) (dual of [8, 8, 1]-code), using
      - discarding factors / shortening the dual code based on linear OA(4⁰, s, F₄, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
        OA with only one run / dual of code without redundancy [i]

(27, 27+6, 48349)-Net over F₄ — Digital

Digital (27, 33, 48349)-net over F₄, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(4³³, 48349, F₄, 6) (dual of [48349, 48316, 7]-code), using
- discarding factors / shortening the dual code based on linear OA(4³³, 65536, F₄, 6) (dual of [65536, 65503, 7]-code), using
  - an extension C_e(5) of the primitive narrow-sense BCH-code C(I) with length 65535Â = 4⁸âˆ’1, defining interval IÂ =Â [1,5], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 6 [i]

(27, 27+6, 2540516)-Net in Base 4 — Upper bound on s

There is no (27, 33, 2540517)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 73 787000 372360 620948 > 4³³ [i]