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

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

(109−5, 109, large)-Net over F₅ — Constructive and digital

Digital (104, 109, large)-net over F₅, using

5² times duplication [i] based on digital (102, 107, large)-net over F₅, using
- t-expansion [i] based on digital (99, 107, 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 (68, 76, 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 (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]

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

There is no (104, 109, large)-net in base 5, because

3 times m-reduction [i] would yield (104, 106, large)-net in base 5, but
- mutually orthogonal hypercube bound [i]

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

(109−5, 109, large)-Net over F5 — Constructive and digital

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

(109−5, 109, large)-Net over F₅ — Constructive and digital