MinT - Best Known (119−10, 119, s)-Nets in Base 5

Best Known (119−10, 119, s)-Nets in Base 5

(119−10, 119, 3371068)-Net over F₅ — Constructive and digital

Digital (109, 119, 3371068)-net over F₅, using

(u,Â u+v)-construction [i] based on
1. digital (22, 27, 15628)-net over F₅, using
  - net defined by OOA [i] based on linear OOA(5²⁷, 15628, F₅, 5, 5) (dual of [(15628, 5), 78113, 6]-NRT-code), using
    - OOA 2-folding and stacking with additional row [i] based on linear OA(5²⁷, 31257, F₅, 5) (dual of [31257, 31230, 6]-code), using
      - 1 times code embedding in larger space [i] based on linear OA(5²⁶, 31256, F₅, 5) (dual of [31256, 31230, 6]-code), using
        trace code [i] based on linear OA(25¹³, 15628, F₂₅, 5) (dual of [15628, 15615, 6]-code), using
        constructionÂ X applied to C_e(4) âŠ‚ C_e(3) [i] based on
        linear OA(25¹³, 15625, F₂₅, 5) (dual of [15625, 15612, 6]-code), using an extension C_e(4) of the primitive narrow-sense BCH-code C(I) with length 15624Â = 25³âˆ’1, defining interval IÂ =Â [1,4], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 5 [i]
        linear OA(25¹⁰, 15625, F₂₅, 4) (dual of [15625, 15615, 5]-code), using an extension C_e(3) of the primitive narrow-sense BCH-code C(I) with length 15624Â = 25³âˆ’1, defining interval IÂ =Â [1,3], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 4 [i]
        linear OA(25⁰, 3, F₂₅, 0) (dual of [3, 3, 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 (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]

(119−10, 119, large)-Net over F₅ — Digital

Digital (109, 119, large)-net over F₅, using

5¹ times duplication [i] based on digital (108, 118, large)-net over F₅, using
- t-expansion [i] based on digital (104, 118, large)-net over F₅, using
  - embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(5¹¹⁸, large, F₅, 14) (dual of [large, large−118, 15]-code), using
    - 7 times code embedding in larger space [i] based on linear OA(5¹¹¹, large, F₅, 14) (dual of [large, large−111, 15]-code), using
      - the primitive expurgated narrow-sense BCH-code C(I) with length 9765624Â = 5¹⁰âˆ’1, defining interval IÂ =Â [0,13], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 15 [i]

(119−10, 119, large)-Net in Base 5 — Upper bound on s

There is no (109, 119, large)-net in base 5, because

8 times m-reduction [i] would yield (109, 111, large)-net in base 5, but
- mutually orthogonal hypercube bound [i]

Best Known (119−10, 119, s)-Nets in Base 5

(119−10, 119, 3371068)-Net over F5 — Constructive and digital

(119−10, 119, large)-Net over F5 — Digital

(119−10, 119, large)-Net in Base 5 — Upper bound on s

(119−10, 119, 3371068)-Net over F₅ — Constructive and digital

(119−10, 119, large)-Net over F₅ — Digital