MinT - Best Known (165−13, 165, s)-Nets in Base 4

Best Known (165−13, 165, s)-Nets in Base 4

(165−13, 165, 5592486)-Net over F₄ — Constructive and digital

Digital (152, 165, 5592486)-net over F₄, using

(u,Â u+v)-construction [i] based on
1. digital (11, 17, 86)-net over F₄, using
  - net defined by OOA [i] based on linear OOA(4¹⁷, 86, F₄, 6, 6) (dual of [(86, 6), 499, 7]-NRT-code), using
    - OA 3-folding and stacking [i] based on linear OA(4¹⁷, 258, F₄, 6) (dual of [258, 241, 7]-code), using
      - discarding factors / shortening the dual code based on linear OA(4¹⁷, 260, F₄, 6) (dual of [260, 243, 7]-code), using
        constructionÂ X applied to C_e(5) âŠ‚ C_e(4) [i] based on
        linear OA(4¹⁷, 256, F₄, 6) (dual of [256, 239, 7]-code), using an extension C_e(5) of the primitive narrow-sense BCH-code C(I) with length 255Â = 4⁴âˆ’1, defining interval IÂ =Â [1,5], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 6 [i]
        linear OA(4¹³, 256, F₄, 5) (dual of [256, 243, 6]-code), using an extension C_e(4) of the primitive narrow-sense BCH-code C(I) with length 255Â = 4⁴âˆ’1, defining interval IÂ =Â [1,4], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 5 [i]
        linear OA(4⁰, 4, F₄, 0) (dual of [4, 4, 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]

(165−13, 165, large)-Net over F₄ — Digital

Digital (152, 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]

(165−13, 165, large)-Net in Base 4 — Upper bound on s

There is no (152, 165, large)-net in base 4, because

11 times m-reduction [i] would yield (152, 154, large)-net in base 4, but
- mutually orthogonal hypercube bound [i]

Best Known (165−13, 165, s)-Nets in Base 4

(165−13, 165, 5592486)-Net over F4 — Constructive and digital

(165−13, 165, large)-Net over F4 — Digital

(165−13, 165, large)-Net in Base 4 — Upper bound on s

(165−13, 165, 5592486)-Net over F₄ — Constructive and digital

(165−13, 165, large)-Net over F₄ — Digital