MinT - Best Known (122−15, 122, s)-Nets in Base 5

Best Known (122−15, 122, s)-Nets in Base 5

Digital (107, 122, 1198371)-net over F₅, using

5² times duplication [i] based on digital (105, 120, 1198371)-net over F₅, using
- net defined by OOA [i] based on linear OOA(5¹²⁰, 1198371, F₅, 15, 15) (dual of [(1198371, 15), 17975445, 16]-NRT-code), using
  - OOA 7-folding and stacking with additional row [i] based on linear OA(5¹²⁰, 8388598, F₅, 15) (dual of [8388598, 8388478, 16]-code), using
    - discarding factors / shortening the dual code 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]

Digital (107, 122, 4540912)-net over F₅, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(5¹²², 4540912, F₅, 15) (dual of [4540912, 4540790, 16]-code), using
- discarding factors / shortening the dual code based on linear OA(5¹²², large, F₅, 15) (dual of [large, large−122, 16]-code), using
  - 2 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]

There is no (107, 122, large)-net in base 5, because

13 times m-reduction [i] would yield (107, 109, large)-net in base 5, but
- mutually orthogonal hypercube bound [i]