MinT - Best Known (109−10, 109, s)-Nets in Base 5

Best Known (109−10, 109, s)-Nets in Base 5

(109−10, 109, 3356071)-Net over F₅ — Constructive and digital

Digital (99, 109, 3356071)-net over F₅, using

(u,Â u+v)-construction [i] based on
1. digital (12, 17, 631)-net over F₅, using
  - net defined by OOA [i] based on linear OOA(5¹⁷, 631, F₅, 6, 5) (dual of [(631, 6), 3769, 6]-NRT-code), using
    - OOA stacking with additional row [i] based on linear OOA(5¹⁷, 632, F₅, 2, 5) (dual of [(632, 2), 1247, 6]-NRT-code), using
      - (u,Â u+v)-construction [i] based on
        linear OOA(5³, 31, F₅, 2, 2) (dual of [(31, 2), 59, 3]-NRT-code), using
        appending kth column [i] based on linear OA(5³, 31, F₅, 2) (dual of [31, 28, 3]-code), using
        Hamming code H(3,5) [i]
        linear OOA(5¹⁴, 601, F₅, 2, 5) (dual of [(601, 2), 1188, 6]-NRT-code), using
        OOA 2-folding [i] based on linear OA(5¹⁴, 1202, F₅, 5) (dual of [1202, 1188, 6]-code), using
        trace code [i] based on linear OA(25⁷, 601, F₂₅, 5) (dual of [601, 594, 6]-code), using
        a code by Danev and Olsson [i]
2. digital (82, 92, 3355440)-net over F₅, using
  - trace code for nets [i] based on digital (36, 46, 1677720)-net over F₂₅, using
    - net defined by OOA [i] based on linear OOA(25⁴⁶, 1677720, F₂₅, 10, 10) (dual of [(1677720, 10), 16777154, 11]-NRT-code), using
      - OA 5-folding and stacking [i] based on linear OA(25⁴⁶, 8388600, F₂₅, 10) (dual of [8388600, 8388554, 11]-code), using
        discarding factors / shortening the dual code based on linear OA(25⁴⁶, large, F₂₅, 10) (dual of [large, large−46, 11]-code), using
        the primitive expurgated narrow-sense BCH-code C(I) with length 9765624Â = 25⁵âˆ’1, defining interval IÂ =Â [0,9], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 11 [i]

(109−10, 109, large)-Net over F₅ — Digital

Digital (99, 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]

(109−10, 109, large)-Net in Base 5 — Upper bound on s

There is no (99, 109, large)-net in base 5, because

8 times m-reduction [i] would yield (99, 101, large)-net in base 5, but
- mutually orthogonal hypercube bound [i]

Best Known (109−10, 109, s)-Nets in Base 5

(109−10, 109, 3356071)-Net over F5 — Constructive and digital

(109−10, 109, large)-Net over F5 — Digital

(109−10, 109, large)-Net in Base 5 — Upper bound on s

(109−10, 109, 3356071)-Net over F₅ — Constructive and digital

(109−10, 109, large)-Net over F₅ — Digital