MinT - Best Known (165−12, 165, s)-Nets in Base 4

Best Known (165−12, 165, s)-Nets in Base 4

(165−12, 165, 5597863)-Net over F₄ — Constructive and digital

Digital (153, 165, 5597863)-net over F₄, using

(u,Â u+v)-construction [i] based on
1. digital (23, 29, 5463)-net over F₄, using
  - net defined by OOA [i] based on linear OOA(4²⁹, 5463, F₄, 6, 6) (dual of [(5463, 6), 32749, 7]-NRT-code), using
    - OA 3-folding and stacking [i] based on linear OA(4²⁹, 16389, F₄, 6) (dual of [16389, 16360, 7]-code), using
      - discarding factors / shortening the dual code based on linear OA(4²⁹, 16391, F₄, 6) (dual of [16391, 16362, 7]-code), using
        constructionÂ X applied to C_e(5) âŠ‚ C_e(4) [i] based on
        linear OA(4²⁹, 16384, F₄, 6) (dual of [16384, 16355, 7]-code), using an extension C_e(5) of the primitive narrow-sense BCH-code C(I) with length 16383Â = 4⁷âˆ’1, defining interval IÂ =Â [1,5], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 6 [i]
        linear OA(4²², 16384, F₄, 5) (dual of [16384, 16362, 6]-code), using an extension C_e(4) of the primitive narrow-sense BCH-code C(I) with length 16383Â = 4⁷âˆ’1, defining interval IÂ =Â [1,4], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 5 [i]
        linear OA(4⁰, 7, F₄, 0) (dual of [7, 7, 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 (124, 136, 5592400)-net over F₄, using
  - trace code for nets [i] based on digital (56, 68, 2796200)-net over F₁₆, using
    - net defined by OOA [i] based on linear OOA(16⁶⁸, 2796200, F₁₆, 14, 12) (dual of [(2796200, 14), 39146732, 13]-NRT-code), using
      - OOA 3-folding and stacking with additional row [i] based on linear OOA(16⁶⁸, 8388601, F₁₆, 2, 12) (dual of [(8388601, 2), 16777134, 13]-NRT-code), using
        discarding factors / shortening the dual code based on linear OOA(16⁶⁸, 8388602, F₁₆, 2, 12) (dual of [(8388602, 2), 16777136, 13]-NRT-code), using
        trace code [i] based on linear OOA(256³⁴, 4194301, F₂₅₆, 2, 12) (dual of [(4194301, 2), 8388568, 13]-NRT-code), using
        OOA 2-folding [i] based on linear OA(256³⁴, 8388602, F₂₅₆, 12) (dual of [8388602, 8388568, 13]-code), using
        discarding factors / shortening the dual code based on linear OA(256³⁴, large, F₂₅₆, 12) (dual of [large, large−34, 13]-code), using
        the primitive expurgated narrow-sense BCH-code C(I) with length 16777215Â = 256³âˆ’1, defining interval IÂ =Â [0,11], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 13 [i]

(165−12, 165, large)-Net over F₄ — Digital

Digital (153, 165, large)-net over F₄, using

t-expansion [i] based on digital (149, 165, large)-net over F₄, using
- 1 times m-reduction [i] based on digital (149, 166, large)-net over F₄, using
  - embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(4¹⁶⁶, large, F₄, 17) (dual of [large, large−166, 18]-code), using
    - 21 times code embedding in larger space [i] based on linear OA(4¹⁴⁵, large, F₄, 17) (dual of [large, large−145, 18]-code), using
      - the expurgated narrow-sense BCH-code C(I) with length 16777217Â | 4²⁴âˆ’1, defining interval IÂ =Â [0,8], and minimum distance dÂ â‰¥ |{−8,−7,…,8}|+1Â =Â 18 (BCH-bound) [i]

(165−12, 165, large)-Net in Base 4 — Upper bound on s

There is no (153, 165, large)-net in base 4, because

10 times m-reduction [i] would yield (153, 155, large)-net in base 4, but
- mutually orthogonal hypercube bound [i]

Best Known (165−12, 165, s)-Nets in Base 4

(165−12, 165, 5597863)-Net over F4 — Constructive and digital

(165−12, 165, large)-Net over F4 — Digital

(165−12, 165, large)-Net in Base 4 — Upper bound on s

(165−12, 165, 5597863)-Net over F₄ — Constructive and digital

(165−12, 165, large)-Net over F₄ — Digital