MinT - Best Known (145−15, 145, s)-Nets in Base 4

Best Known (145−15, 145, s)-Nets in Base 4

(145−15, 145, 1198391)-Net over F₄ — Constructive and digital

Digital (130, 145, 1198391)-net over F₄, using

4¹ times duplication [i] based on digital (129, 144, 1198391)-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 (118, 133, 1198371)-net over F₄, using
    - net defined by OOA [i] based on linear OOA(4¹³³, 1198371, F₄, 15, 15) (dual of [(1198371, 15), 17975432, 16]-NRT-code), using
      - OOA 7-folding and stacking with additional row [i] based on linear OA(4¹³³, 8388598, F₄, 15) (dual of [8388598, 8388465, 16]-code), using
        discarding factors / shortening the dual code based on linear OA(4¹³³, large, F₄, 15) (dual of [large, large−133, 16]-code), using
        the primitive expurgated narrow-sense BCH-code C(I) with length 16777215Â = 4¹²âˆ’1, defining interval IÂ =Â [0,14], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 16 [i]

(145−15, 145, large)-Net over F₄ — Digital

Digital (130, 145, large)-net over F₄, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(4¹⁴⁵, large, F₄, 15) (dual of [large, large−145, 16]-code), using
- strength reduction [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]

(145−15, 145, large)-Net in Base 4 — Upper bound on s

There is no (130, 145, large)-net in base 4, because

13 times m-reduction [i] would yield (130, 132, large)-net in base 4, but
- mutually orthogonal hypercube bound [i]