MinT - Best Known (139−6, 139, s)-Nets in Base 5

Best Known (139−6, 139, s)-Nets in Base 5

(139−6, 139, large)-Net over F₅ — Constructive and digital

Digital (133, 139, large)-net over F₅, using

5²² times duplication [i] based on digital (111, 117, large)-net over F₅, using
- t-expansion [i] based on digital (108, 117, large)-net over F₅, using
  - (u,Â u+v)-construction [i] based on
    1. digital (27, 31, 4194301)-net over F₅, using
      - net defined by OOA [i] based on linear OOA(5³¹, 4194301, F₅, 4, 4) (dual of [(4194301, 4), 16777173, 5]-NRT-code), using
        appending kth column [i] based on linear OOA(5³¹, 4194301, F₅, 3, 4) (dual of [(4194301, 3), 12582872, 5]-NRT-code), using
        OA 2-folding and stacking [i] based on linear OA(5³¹, 8388602, F₅, 4) (dual of [8388602, 8388571, 5]-code), using
        discarding factors / shortening the dual code based on linear OA(5³¹, large, F₅, 4) (dual of [large, large−31, 5]-code), using
        the primitive expurgated narrow-sense BCH-code C(I) with length 9765624Â = 5¹⁰âˆ’1, defining interval IÂ =Â [0,3], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 5 [i]
    2. digital (77, 86, 4194306)-net over F₅, using
      - (u,Â u+v)-construction [i] based on
        digital (0, 4, 6)-net over F₅, using
        net from sequence [i] based on digital (0, 5)-sequence over F₅, using
        Faure sequence [i]
        Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₅ with g(F) = 0 and N(F) ≥ 6, using
        the rational function field F₅(x) [i]
        Niederreiter sequence [i]
        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]

(139−6, 139, large)-Net in Base 5 — Upper bound on s

There is no (133, 139, large)-net in base 5, because

4 times m-reduction [i] would yield (133, 135, large)-net in base 5, but
- mutually orthogonal hypercube bound [i]

Best Known (139−6, 139, s)-Nets in Base 5

(139−6, 139, large)-Net over F5 — Constructive and digital

(139−6, 139, large)-Net in Base 5 — Upper bound on s

(139−6, 139, large)-Net over F₅ — Constructive and digital