MinT - Best Known (143−14, 143, s)-Nets in Base 5

Best Known (143−14, 143, s)-Nets in Base 5

(143−14, 143, 2396767)-Net over F₅ — Constructive and digital

Digital (129, 143, 2396767)-net over F₅, using

5¹ times duplication [i] based on digital (128, 142, 2396767)-net over F₅, using
- (u,Â u+v)-construction [i] based on
  1. digital (3, 10, 25)-net over F₅, using
    - linear code embedding found in a computer search [i]
  2. digital (118, 132, 2396742)-net over F₅, using
    - trace code for nets [i] based on digital (52, 66, 1198371)-net over F₂₅, using
      - net defined by OOA [i] based on linear OOA(25⁶⁶, 1198371, F₂₅, 14, 14) (dual of [(1198371, 14), 16777128, 15]-NRT-code), using
        OA 7-folding and stacking [i] based on linear OA(25⁶⁶, 8388597, F₂₅, 14) (dual of [8388597, 8388531, 15]-code), using
        discarding factors / shortening the dual code based on linear OA(25⁶⁶, large, F₂₅, 14) (dual of [large, large−66, 15]-code), using
        the primitive expurgated narrow-sense BCH-code C(I) with length 9765624Â = 25⁵âˆ’1, defining interval IÂ =Â [0,13], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 15 [i]

(143−14, 143, large)-Net over F₅ — Digital

Digital (129, 143, large)-net over F₅, using

3 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]

(143−14, 143, large)-Net in Base 5 — Upper bound on s

There is no (129, 143, large)-net in base 5, because

12 times m-reduction [i] would yield (129, 131, large)-net in base 5, but
- mutually orthogonal hypercube bound [i]