MinT - Best Known (99−10, 99, s)-Nets in Base 5

Best Known (99−10, 99, s)-Nets in Base 5

(99−10, 99, 3355461)-Net over F₅ — Constructive and digital

Digital (89, 99, 3355461)-net over F₅, using

(u,Â u+v)-construction [i] based on
1. digital (2, 7, 21)-net over F₅, using
  - linear code embedding found in a computer search [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]

(99−10, 99, large)-Net over F₅ — Digital

Digital (89, 99, large)-net over F₅, using

t-expansion [i] based on digital (88, 99, large)-net over F₅, using
- 1 times m-reduction [i] based on digital (88, 100, large)-net over F₅, using
  - embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(5¹⁰⁰, large, F₅, 12) (dual of [large, large−100, 13]-code), using
    - 9 times code embedding in larger space [i] based on linear OA(5⁹¹, large, F₅, 12) (dual of [large, large−91, 13]-code), using
      - the primitive expurgated narrow-sense BCH-code C(I) with length 9765624Â = 5¹⁰âˆ’1, defining interval IÂ =Â [0,11], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 13 [i]

(99−10, 99, large)-Net in Base 5 — Upper bound on s

There is no (89, 99, large)-net in base 5, because

8 times m-reduction [i] would yield (89, 91, large)-net in base 5, but
- mutually orthogonal hypercube bound [i]