MinT - Best Known (15−6, 15, s)-Nets in Base 49

Best Known (15−6, 15, s)-Nets in Base 49

(15−6, 15, 2451)-Net over F₄₉ — Constructive and digital

Digital (9, 15, 2451)-net over F₄₉, using

net defined by OOA [i] based on linear OOA(49¹⁵, 2451, F₄₉, 6, 6) (dual of [(2451, 6), 14691, 7]-NRT-code), using
- OA 3-folding and stacking [i] based on linear OA(49¹⁵, 7353, F₄₉, 6) (dual of [7353, 7338, 7]-code), using
  - the BCH-code C(I) with length 7353Â | 49³âˆ’1, defining interval IÂ =Â {1,17,33,…,81}, and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 7 [i]

(15−6, 15, 7353)-Net over F₄₉ — Digital

Digital (9, 15, 7353)-net over F₄₉, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(49¹⁵, 7353, F₄₉, 6) (dual of [7353, 7338, 7]-code), using
- the BCH-code C(I) with length 7353Â | 49³âˆ’1, defining interval IÂ =Â {1,17,33,…,81}, and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 7 [i]

(15−6, 15, large)-Net in Base 49 — Upper bound on s

There is no (9, 15, large)-net in base 49, because

4 times m-reduction [i] would yield (9, 11, large)-net in base 49, but
- mutually orthogonal hypercube bound [i]