MinT - Best Known (42, 42+12, s)-Nets in Base 5

Best Known (42, 42+12, s)-Nets in Base 5

(42, 42+12, 651)-Net over F₅ — Constructive and digital

Digital (42, 54, 651)-net over F₅, using

net defined by OOA [i] based on linear OOA(5⁵⁴, 651, F₅, 12, 12) (dual of [(651, 12), 7758, 13]-NRT-code), using
- OA 6-folding and stacking [i] based on linear OA(5⁵⁴, 3906, F₅, 12) (dual of [3906, 3852, 13]-code), using
  - the BCH-code C(I) with length 3906Â | 5⁶âˆ’1, defining interval IÂ =Â {−394,−3,388,…,1}, and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 13 [i]

(42, 42+12, 3906)-Net over F₅ — Digital

Digital (42, 54, 3906)-net over F₅, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(5⁵⁴, 3906, F₅, 12) (dual of [3906, 3852, 13]-code), using
- the BCH-code C(I) with length 3906Â | 5⁶âˆ’1, defining interval IÂ =Â {−394,−3,388,…,1}, and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 13 [i]

(42, 42+12, 1461810)-Net in Base 5 — Upper bound on s

There is no (42, 54, 1461811)-net in base 5, because

the generalized Rao bound for nets shows that 5^mÂ â‰¥ 55 511368 670817 268527 025284 000984 110825 > 5⁵⁴ [i]