MinT - Best Known (109−9, 109, s)-Nets in Base 5

Best Known (109−9, 109, s)-Nets in Base 5

(109−9, 109, 4584929)-Net over F₅ — Constructive and digital

Digital (100, 109, 4584929)-net over F₅, using

5¹ times duplication [i] based on digital (99, 108, 4584929)-net over F₅, using
- (u,Â u+v)-construction [i] based on
  1. digital (22, 26, 390629)-net over F₅, using
    - net defined by OOA [i] based on linear OOA(5²⁶, 390629, F₅, 4, 4) (dual of [(390629, 4), 1562490, 5]-NRT-code), using
      - appending kth column [i] based on linear OOA(5²⁶, 390629, F₅, 3, 4) (dual of [(390629, 3), 1171861, 5]-NRT-code), using
        OA 2-folding and stacking [i] based on linear OA(5²⁶, 781258, F₅, 4) (dual of [781258, 781232, 5]-code), using
        trace code [i] based on linear OA(25¹³, 390629, F₂₅, 4) (dual of [390629, 390616, 5]-code), using
        constructionÂ X applied to C_e(3) âŠ‚ C_e(2) [i] based on
        linear OA(25¹³, 390625, F₂₅, 4) (dual of [390625, 390612, 5]-code), using an extension C_e(3) of the primitive narrow-sense BCH-code C(I) with length 390624Â = 25⁴âˆ’1, defining interval IÂ =Â [1,3], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 4 [i]
        linear OA(25⁹, 390625, F₂₅, 3) (dual of [390625, 390616, 4]-code or 390625-cap in PG(8,25)), using an extension C_e(2) of the primitive narrow-sense BCH-code C(I) with length 390624Â = 25⁴âˆ’1, defining interval IÂ =Â [1,2], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 3 [i]
        linear OA(25⁰, 4, F₂₅, 0) (dual of [4, 4, 1]-code), using
        discarding factors / shortening the dual code based on linear OA(25⁰, s, F₂₅, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
        OA with only one run / dual of code without redundancy [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]

(109−9, 109, large)-Net over F₅ — Digital

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]

(109−9, 109, large)-Net in Base 5 — Upper bound on s

There is no (100, 109, large)-net in base 5, because

7 times m-reduction [i] would yield (100, 102, large)-net in base 5, but
- mutually orthogonal hypercube bound [i]

Best Known (109−9, 109, s)-Nets in Base 5

(109−9, 109, 4584929)-Net over F5 — Constructive and digital

(109−9, 109, large)-Net over F5 — Digital

(109−9, 109, large)-Net in Base 5 — Upper bound on s

(109−9, 109, 4584929)-Net over F₅ — Constructive and digital

(109−9, 109, large)-Net over F₅ — Digital