MinT - Best Known (255−17, 255, s)-Nets in Base 4

Best Known (255−17, 255, s)-Nets in Base 4

(255−17, 255, 4259846)-Net over F₄ — Constructive and digital

Digital (238, 255, 4259846)-net over F₄, using

(u,Â u+v)-construction [i] based on
1. digital (51, 59, 65546)-net over F₄, using
  - (u,Â u+v)-construction [i] based on
    1. digital (1, 5, 10)-net over F₄, using
      - linear code embedding found in a computer search [i]
    2. digital (46, 54, 65536)-net over F₄, using
      - net defined by OOA [i] based on linear OOA(4⁵⁴, 65536, F₄, 8, 8) (dual of [(65536, 8), 524234, 9]-NRT-code), using
        OA 4-folding and stacking [i] based on linear OA(4⁵⁴, 262144, F₄, 8) (dual of [262144, 262090, 9]-code), using
        1 times truncation [i] based on linear OA(4⁵⁵, 262145, F₄, 9) (dual of [262145, 262090, 10]-code), using
        the expurgated narrow-sense BCH-code C(I) with length 262145Â | 4¹⁸âˆ’1, defining interval IÂ =Â [0,4], and minimum distance dÂ â‰¥ |{−4,−3,…,4}|+1Â =Â 10 (BCH-bound) [i]
2. digital (179, 196, 4194300)-net over F₄, using
  - trace code for nets [i] based on digital (81, 98, 2097150)-net over F₁₆, using
    - net defined by OOA [i] based on linear OOA(16⁹⁸, 2097150, F₁₆, 18, 17) (dual of [(2097150, 18), 37748602, 18]-NRT-code), using
      - OOA 4-folding and stacking with additional row [i] based on linear OOA(16⁹⁸, 8388601, F₁₆, 2, 17) (dual of [(8388601, 2), 16777104, 18]-NRT-code), using
        discarding factors / shortening the dual code based on linear OOA(16⁹⁸, 8388602, F₁₆, 2, 17) (dual of [(8388602, 2), 16777106, 18]-NRT-code), using
        trace code [i] based on linear OOA(256⁴⁹, 4194301, F₂₅₆, 2, 17) (dual of [(4194301, 2), 8388553, 18]-NRT-code), using
        OOA 2-folding [i] based on linear OA(256⁴⁹, 8388602, F₂₅₆, 17) (dual of [8388602, 8388553, 18]-code), using
        discarding factors / shortening the dual code based on linear OA(256⁴⁹, large, F₂₅₆, 17) (dual of [large, large−49, 18]-code), using
        the primitive expurgated narrow-sense BCH-code C(I) with length 16777215Â = 256³âˆ’1, defining interval IÂ =Â [0,16], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 18 [i]

(255−17, 255, large)-Net over F₄ — Digital

Digital (238, 255, large)-net over F₄, using

t-expansion [i] based on digital (231, 255, large)-net over F₄, using
- 2 times m-reduction [i] based on digital (231, 257, large)-net over F₄, using
  - embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(4²⁵⁷, large, F₄, 26) (dual of [large, large−257, 27]-code), using
    - 28 times code embedding in larger space [i] based on linear OA(4²²⁹, large, F₄, 26) (dual of [large, large−229, 27]-code), using
      - the primitive expurgated narrow-sense BCH-code C(I) with length 16777215Â = 4¹²âˆ’1, defining interval IÂ =Â [0,25], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 27 [i]

(255−17, 255, large)-Net in Base 4 — Upper bound on s

There is no (238, 255, large)-net in base 4, because

15 times m-reduction [i] would yield (238, 240, large)-net in base 4, but
- mutually orthogonal hypercube bound [i]

Best Known (255−17, 255, s)-Nets in Base 4

(255−17, 255, 4259846)-Net over F4 — Constructive and digital

(255−17, 255, large)-Net over F4 — Digital

(255−17, 255, large)-Net in Base 4 — Upper bound on s

(255−17, 255, 4259846)-Net over F₄ — Constructive and digital

(255−17, 255, large)-Net over F₄ — Digital