MinT - Best Known (20−6, 20, s)-Nets in Base 5

Best Known (20−6, 20, s)-Nets in Base 5

(20−6, 20, 521)-Net over F₅ — Constructive and digital

Digital (14, 20, 521)-net over F₅, using

net defined by OOA [i] based on linear OOA(5²⁰, 521, F₅, 6, 6) (dual of [(521, 6), 3106, 7]-NRT-code), using
- OA 3-folding and stacking [i] based on linear OA(5²⁰, 1563, F₅, 6) (dual of [1563, 1543, 7]-code), using
  - the BCH-code C(I) with length 1563Â | 5¹⁰âˆ’1, defining interval IÂ =Â {−5,−3,−1,…,5}, and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 7 [i]

(20−6, 20, 1154)-Net over F₅ — Digital

Digital (14, 20, 1154)-net over F₅, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(5²⁰, 1154, F₅, 6) (dual of [1154, 1134, 7]-code), using
- discarding factors / shortening the dual code based on linear OA(5²⁰, 1563, F₅, 6) (dual of [1563, 1543, 7]-code), using
  - the BCH-code C(I) with length 1563Â | 5¹⁰âˆ’1, defining interval IÂ =Â {−5,−3,−1,…,5}, and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 7 [i]

(20−6, 20, 20753)-Net in Base 5 — Upper bound on s

There is no (14, 20, 20754)-net in base 5, because

the generalized Rao bound for nets shows that 5^mÂ â‰¥ 95 377077 944809 > 5²⁰ [i]