MinT - Best Known (111−9, 111, s)-Nets in Base 5

Best Known (111−9, 111, s)-Nets in Base 5

(111−9, 111, 5170872)-Net over F₅ — Constructive and digital

Digital (102, 111, 5170872)-net over F₅, using

(u,Â u+v)-construction [i] based on
1. digital (25, 29, 976572)-net over F₅, using
  - net defined by OOA [i] based on linear OOA(5²⁹, 976572, F₅, 4, 4) (dual of [(976572, 4), 3906259, 5]-NRT-code), using
    - appending kth column [i] based on linear OOA(5²⁹, 976572, F₅, 3, 4) (dual of [(976572, 3), 2929687, 5]-NRT-code), using
      - OA 2-folding and stacking [i] based on linear OA(5²⁹, 1953144, F₅, 4) (dual of [1953144, 1953115, 5]-code), using
        constructionÂ X applied to C_e(3) âŠ‚ C_e(1) [i] based on
        linear OA(5²⁸, 1953125, F₅, 4) (dual of [1953125, 1953097, 5]-code), using an extension C_e(3) of the primitive narrow-sense BCH-code C(I) with length 1953124Â = 5⁹âˆ’1, defining interval IÂ =Â [1,3], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 4 [i]
        linear OA(5¹⁰, 1953125, F₅, 2) (dual of [1953125, 1953115, 3]-code), using an extension C_e(1) of the primitive narrow-sense BCH-code C(I) with length 1953124Â = 5⁹âˆ’1, defining interval IÂ =Â [1,1], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 2 [i]
        linear OA(5¹, 19, F₅, 1) (dual of [19, 18, 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 (73, 82, 4194300)-net over F₅, using
  - net defined by OOA [i] based on linear OOA(5⁸², 4194300, F₅, 10, 9) (dual of [(4194300, 10), 41942918, 10]-NRT-code), using
    - OOA 2-folding and stacking with additional row [i] based on linear OOA(5⁸², 8388601, F₅, 2, 9) (dual of [(8388601, 2), 16777120, 10]-NRT-code), using
      - discarding factors / shortening the dual code based on linear OOA(5⁸², 8388602, F₅, 2, 9) (dual of [(8388602, 2), 16777122, 10]-NRT-code), using
        trace code [i] based on linear OOA(25⁴¹, 4194301, F₂₅, 2, 9) (dual of [(4194301, 2), 8388561, 10]-NRT-code), using
        OOA 2-folding [i] based on linear OA(25⁴¹, 8388602, F₂₅, 9) (dual of [8388602, 8388561, 10]-code), using
        discarding factors / shortening the dual code based on linear OA(25⁴¹, large, F₂₅, 9) (dual of [large, large−41, 10]-code), using
        the expurgated narrow-sense BCH-code C(I) with length 9765626Â | 25¹⁰âˆ’1, defining interval IÂ =Â [0,4], and minimum distance dÂ â‰¥ |{−4,−3,…,4}|+1Â =Â 10 (BCH-bound) [i]

(111−9, 111, large)-Net over F₅ — Digital

Digital (102, 111, large)-net over F₅, using

5² times duplication [i] based on digital (100, 109, large)-net over F₅, using
- t-expansion [i] based on digital (96, 109, large)-net over F₅, using
  - embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(5¹⁰⁹, large, F₅, 13) (dual of [large, large−109, 14]-code), using
    - 8 times code embedding in larger space [i] based on linear OA(5¹⁰¹, large, F₅, 13) (dual of [large, large−101, 14]-code), using
      - the expurgated narrow-sense BCH-code C(I) with length 9765626Â | 5²⁰âˆ’1, defining interval IÂ =Â [0,6], and minimum distance dÂ â‰¥ |{−6,−5,…,6}|+1Â =Â 14 (BCH-bound) [i]

(111−9, 111, large)-Net in Base 5 — Upper bound on s

There is no (102, 111, large)-net in base 5, because

7 times m-reduction [i] would yield (102, 104, large)-net in base 5, but
- mutually orthogonal hypercube bound [i]

Best Known (111−9, 111, s)-Nets in Base 5

(111−9, 111, 5170872)-Net over F5 — Constructive and digital

(111−9, 111, large)-Net over F5 — Digital

(111−9, 111, large)-Net in Base 5 — Upper bound on s

(111−9, 111, 5170872)-Net over F₅ — Constructive and digital

(111−9, 111, large)-Net over F₅ — Digital