MinT - Best Known (145−10, 145, s)-Nets in Base 5

Best Known (145−10, 145, s)-Nets in Base 5

(145−10, 145, 7148981)-Net over F₅ — Constructive and digital

Digital (135, 145, 7148981)-net over F₅, using

generalized (u,Â u+v)-construction [i] based on
1. digital (10, 13, 438101)-net over F₅, using
  - net defined by OOA [i] based on linear OOA(5¹³, 438101, F₅, 3, 3) (dual of [(438101, 3), 1314290, 4]-NRT-code), using
    - appending kth column [i] based on linear OOA(5¹³, 438101, F₅, 2, 3) (dual of [(438101, 2), 876189, 4]-NRT-code), using
      - OAs with strength 3, bÂ â‰ Â 2, and mÂ >Â 3 are always embeddable [i] based on linear OA(5¹³, 438101, F₅, 3) (dual of [438101, 438088, 4]-code or 438101-cap in PG(12,5)), using
        product of projective cap with cap in AG(6,Â 5) [i] based on linear OA(5⁷, 675, F₅, 3) (dual of [675, 668, 4]-code or 675-cap in PG(6,5)), using
        product of two caps with tangent hyperplane [i]
2. digital (35, 40, 3355440)-net over F₅, using
  - s-reduction based on digital (35, 40, 4194301)-net over F₅, using
    - net defined by OOA [i] based on linear OOA(5⁴⁰, 4194301, F₅, 5, 5) (dual of [(4194301, 5), 20971465, 6]-NRT-code), using
      - OOA 2-folding and stacking with additional row [i] based on linear OA(5⁴⁰, large, F₅, 5) (dual of [large, large−40, 6]-code), using
        the primitive narrow-sense BCH-code C(I) with length 9765624Â = 5¹⁰âˆ’1, defining interval IÂ =Â [1,5], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 6 [i]
3. 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]

(145−10, 145, large)-Net over F₅ — Digital

Digital (135, 145, large)-net over F₅, using

t-expansion [i] based on digital (129, 145, large)-net over F₅, using
- 1 times m-reduction [i] based on digital (129, 146, large)-net over F₅, using
  - embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(5¹⁴⁶, large, F₅, 17) (dual of [large, large−146, 18]-code), using
    - 15 times code embedding in larger space [i] based on linear OA(5¹³¹, large, F₅, 17) (dual of [large, large−131, 18]-code), using
      - the primitive expurgated narrow-sense BCH-code C(I) with length 9765624Â = 5¹⁰âˆ’1, defining interval IÂ =Â [0,16], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 18 [i]

(145−10, 145, large)-Net in Base 5 — Upper bound on s

There is no (135, 145, large)-net in base 5, because

8 times m-reduction [i] would yield (135, 137, large)-net in base 5, but
- mutually orthogonal hypercube bound [i]

Best Known (145−10, 145, s)-Nets in Base 5

(145−10, 145, 7148981)-Net over F5 — Constructive and digital

(145−10, 145, large)-Net over F5 — Digital

(145−10, 145, large)-Net in Base 5 — Upper bound on s

(145−10, 145, 7148981)-Net over F₅ — Constructive and digital

(145−10, 145, large)-Net over F₅ — Digital