MinT - Best Known (89−8, 89, s)-Nets in Base 5

Best Known (89−8, 89, s)-Nets in Base 5

(89−8, 89, 4195868)-Net over F₅ — Constructive and digital

Digital (81, 89, 4195868)-net over F₅, using

(u,Â u+v)-construction [i] based on
1. digital (13, 17, 1568)-net over F₅, using
  - net defined by OOA [i] based on linear OOA(5¹⁷, 1568, F₅, 4, 4) (dual of [(1568, 4), 6255, 5]-NRT-code), using
    - appending kth column [i] based on linear OOA(5¹⁷, 1568, F₅, 3, 4) (dual of [(1568, 3), 4687, 5]-NRT-code), using
      - OA 2-folding and stacking [i] based on linear OA(5¹⁷, 3136, F₅, 4) (dual of [3136, 3119, 5]-code), using
        constructionÂ X applied to C_e(3) âŠ‚ C_e(1) [i] based on
        linear OA(5¹⁶, 3125, F₅, 4) (dual of [3125, 3109, 5]-code), using an extension C_e(3) of the primitive narrow-sense BCH-code C(I) with length 3124Â = 5⁵âˆ’1, defining interval IÂ =Â [1,3], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 4 [i]
        linear OA(5⁶, 3125, F₅, 2) (dual of [3125, 3119, 3]-code), using an extension C_e(1) of the primitive narrow-sense BCH-code C(I) with length 3124Â = 5⁵âˆ’1, defining interval IÂ =Â [1,1], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 2 [i]
        linear OA(5¹, 11, F₅, 1) (dual of [11, 10, 2]-code), using
        discarding factors / shortening the dual code based on linear OA(5¹, s, F₅, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
        parity-check code [i]
2. digital (64, 72, 4194300)-net over F₅, using
  - net defined by OOA [i] based on linear OOA(5⁷², 4194300, F₅, 10, 8) (dual of [(4194300, 10), 41942928, 9]-NRT-code), using
    - OOA 2-folding and stacking with additional row [i] based on linear OOA(5⁷², 8388601, F₅, 2, 8) (dual of [(8388601, 2), 16777130, 9]-NRT-code), using
      - discarding factors / shortening the dual code based on linear OOA(5⁷², 8388602, F₅, 2, 8) (dual of [(8388602, 2), 16777132, 9]-NRT-code), using
        trace code [i] based on linear OOA(25³⁶, 4194301, F₂₅, 2, 8) (dual of [(4194301, 2), 8388566, 9]-NRT-code), using
        OOA 2-folding [i] based on linear OA(25³⁶, 8388602, F₂₅, 8) (dual of [8388602, 8388566, 9]-code), using
        discarding factors / shortening the dual code based on linear OA(25³⁶, large, F₂₅, 8) (dual of [large, large−36, 9]-code), using
        the primitive expurgated narrow-sense BCH-code C(I) with length 9765624Â = 25⁵âˆ’1, defining interval IÂ =Â [0,7], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 9 [i]

(89−8, 89, large)-Net over F₅ — Digital

Digital (81, 89, large)-net over F₅, using

t-expansion [i] based on digital (79, 89, large)-net over F₅, using
- 1 times m-reduction [i] based on digital (79, 90, large)-net over F₅, using
  - embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(5⁹⁰, large, F₅, 11) (dual of [large, large−90, 12]-code), using
    - 9 times code embedding in larger space [i] based on linear OA(5⁸¹, large, F₅, 11) (dual of [large, large−81, 12]-code), using
      - the expurgated narrow-sense BCH-code C(I) with length 9765626Â | 5²⁰âˆ’1, defining interval IÂ =Â [0,5], and minimum distance dÂ â‰¥ |{−5,−4,…,5}|+1Â =Â 12 (BCH-bound) [i]

(89−8, 89, large)-Net in Base 5 — Upper bound on s

There is no (81, 89, large)-net in base 5, because

6 times m-reduction [i] would yield (81, 83, large)-net in base 5, but
- mutually orthogonal hypercube bound [i]

Best Known (89−8, 89, s)-Nets in Base 5

(89−8, 89, 4195868)-Net over F5 — Constructive and digital

(89−8, 89, large)-Net over F5 — Digital

(89−8, 89, large)-Net in Base 5 — Upper bound on s

(89−8, 89, 4195868)-Net over F₅ — Constructive and digital

(89−8, 89, large)-Net over F₅ — Digital