MinT - Best Known (123−10, 123, s)-Nets in Base 5

Best Known (123−10, 123, s)-Nets in Base 5

(123−10, 123, 3394532)-Net over F₅ — Constructive and digital

Digital (113, 123, 3394532)-net over F₅, using

(u,Â u+v)-construction [i] based on
1. digital (26, 31, 39092)-net over F₅, using
  - net defined by OOA [i] based on linear OOA(5³¹, 39092, F₅, 6, 5) (dual of [(39092, 6), 234521, 6]-NRT-code), using
    - OOA stacking with additional row [i] based on linear OOA(5³¹, 39093, F₅, 2, 5) (dual of [(39093, 2), 78155, 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(5²⁸, 39062, F₅, 2, 5) (dual of [(39062, 2), 78096, 6]-NRT-code), using
        OOA 2-folding [i] based on linear OA(5²⁸, 78124, F₅, 5) (dual of [78124, 78096, 6]-code), using
        1 times truncation [i] based on linear OA(5²⁹, 78125, F₅, 6) (dual of [78125, 78096, 7]-code), using
        an extension C_e(5) of the primitive narrow-sense BCH-code C(I) with length 78124Â = 5⁷âˆ’1, defining interval IÂ =Â [1,5], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 6 [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]

(123−10, 123, large)-Net over F₅ — Digital

Digital (113, 123, large)-net over F₅, using

t-expansion [i] based on digital (112, 123, large)-net over F₅, using
- 4 times m-reduction [i] based on digital (112, 127, large)-net over F₅, using
  - embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(5¹²⁷, large, F₅, 15) (dual of [large, large−127, 16]-code), using
    - 7 times code embedding in larger space [i] based on linear OA(5¹²⁰, large, F₅, 15) (dual of [large, large−120, 16]-code), using
      - the primitive narrow-sense BCH-code C(I) with length 9765624Â = 5¹⁰âˆ’1, defining interval IÂ =Â [1,15], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 16 [i]

(123−10, 123, large)-Net in Base 5 — Upper bound on s

There is no (113, 123, large)-net in base 5, because

8 times m-reduction [i] would yield (113, 115, large)-net in base 5, but
- mutually orthogonal hypercube bound [i]

Best Known (123−10, 123, s)-Nets in Base 5

(123−10, 123, 3394532)-Net over F5 — Constructive and digital

(123−10, 123, large)-Net over F5 — Digital

(123−10, 123, large)-Net in Base 5 — Upper bound on s

(123−10, 123, 3394532)-Net over F₅ — Constructive and digital

(123−10, 123, large)-Net over F₅ — Digital