MinT - Best Known (181−13, 181, s)-Nets in Base 4

Best Known (181−13, 181, s)-Nets in Base 4

(181−13, 181, 5614248)-Net over F₄ — Constructive and digital

Digital (168, 181, 5614248)-net over F₄, using

(u,Â u+v)-construction [i] based on
1. 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
        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]
        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]
        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]
2. digital (135, 148, 5592400)-net over F₄, using
  - trace code for nets [i] based on digital (61, 74, 2796200)-net over F₁₆, using
    - net defined by OOA [i] based on linear OOA(16⁷⁴, 2796200, F₁₆, 14, 13) (dual of [(2796200, 14), 39146726, 14]-NRT-code), using
      - OOA 3-folding and stacking with additional row [i] based on linear OOA(16⁷⁴, 8388601, F₁₆, 2, 13) (dual of [(8388601, 2), 16777128, 14]-NRT-code), using
        discarding factors / shortening the dual code based on linear OOA(16⁷⁴, 8388602, F₁₆, 2, 13) (dual of [(8388602, 2), 16777130, 14]-NRT-code), using
        trace code [i] based on linear OOA(256³⁷, 4194301, F₂₅₆, 2, 13) (dual of [(4194301, 2), 8388565, 14]-NRT-code), using
        OOA 2-folding [i] based on linear OA(256³⁷, 8388602, F₂₅₆, 13) (dual of [8388602, 8388565, 14]-code), using
        discarding factors / shortening the dual code based on linear OA(256³⁷, large, F₂₅₆, 13) (dual of [large, large−37, 14]-code), using
        the primitive expurgated narrow-sense BCH-code C(I) with length 16777215Â = 256³âˆ’1, defining interval IÂ =Â [0,12], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 14 [i]

(181−13, 181, large)-Net over F₄ — Digital

Digital (168, 181, large)-net over F₄, using

t-expansion [i] based on digital (167, 181, large)-net over F₄, using
- 5 times m-reduction [i] based on digital (167, 186, large)-net over F₄, using
  - embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(4¹⁸⁶, large, F₄, 19) (dual of [large, large−186, 20]-code), using
    - 17 times code embedding in larger space [i] based on linear OA(4¹⁶⁹, large, F₄, 19) (dual of [large, large−169, 20]-code), using
      - the expurgated narrow-sense BCH-code C(I) with length 16777217Â | 4²⁴âˆ’1, defining interval IÂ =Â [0,9], and minimum distance dÂ â‰¥ |{−9,−8,…,9}|+1Â =Â 20 (BCH-bound) [i]

(181−13, 181, large)-Net in Base 4 — Upper bound on s

There is no (168, 181, large)-net in base 4, because

11 times m-reduction [i] would yield (168, 170, large)-net in base 4, but
- mutually orthogonal hypercube bound [i]

Best Known (181−13, 181, s)-Nets in Base 4

(181−13, 181, 5614248)-Net over F4 — Constructive and digital

(181−13, 181, large)-Net over F4 — Digital

(181−13, 181, large)-Net in Base 4 — Upper bound on s

(181−13, 181, 5614248)-Net over F₄ — Constructive and digital

(181−13, 181, large)-Net over F₄ — Digital