MinT - Best Known (172−14, 172, s)-Nets in Base 4

Best Known (172−14, 172, s)-Nets in Base 4

(172−14, 172, 4793504)-Net over F₄ — Constructive and digital

Digital (158, 172, 4793504)-net over F₄, using

4¹ times duplication [i] based on digital (157, 171, 4793504)-net over F₄, using
- (u,Â u+v)-construction [i] based on
  1. digital (4, 11, 20)-net over F₄, using
    - linear code embedding found in a computer search [i]
  2. digital (146, 160, 4793484)-net over F₄, using
    - trace code for nets [i] based on digital (26, 40, 1198371)-net over F₂₅₆, using
      - net defined by OOA [i] based on linear OOA(256⁴⁰, 1198371, F₂₅₆, 14, 14) (dual of [(1198371, 14), 16777154, 15]-NRT-code), using
        OA 7-folding and stacking [i] based on linear OA(256⁴⁰, 8388597, F₂₅₆, 14) (dual of [8388597, 8388557, 15]-code), using
        discarding factors / shortening the dual code based on linear OA(256⁴⁰, large, F₂₅₆, 14) (dual of [large, large−40, 15]-code), using
        the primitive expurgated narrow-sense BCH-code C(I) with length 16777215Â = 256³âˆ’1, defining interval IÂ =Â [0,13], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 15 [i]

(172−14, 172, large)-Net over F₄ — Digital

Digital (158, 172, large)-net over F₄, using

4 times m-reduction [i] based on digital (158, 176, large)-net over F₄, using
- embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(4¹⁷⁶, large, F₄, 18) (dual of [large, large−176, 19]-code), using
  - 19 times code embedding in larger space [i] based on linear OA(4¹⁵⁷, large, F₄, 18) (dual of [large, large−157, 19]-code), using
    - the primitive expurgated narrow-sense BCH-code C(I) with length 16777215Â = 4¹²âˆ’1, defining interval IÂ =Â [0,17], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 19 [i]

(172−14, 172, large)-Net in Base 4 — Upper bound on s

There is no (158, 172, large)-net in base 4, because

12 times m-reduction [i] would yield (158, 160, large)-net in base 4, but
- mutually orthogonal hypercube bound [i]