Best Known (52, 52+7, s)-Nets in Base 5
(52, 52+7, 2797915)-Net over F5 — Constructive and digital
Digital (52, 59, 2797915)-net over F5, using
- net defined by OOA [i] based on linear OOA(559, 2797915, F5, 9, 7) (dual of [(2797915, 9), 25181176, 8]-NRT-code), using
- OOA stacking with additional row [i] based on linear OOA(559, 2797916, F5, 3, 7) (dual of [(2797916, 3), 8393689, 8]-NRT-code), using
- (u, u+v)-construction [i] based on
- linear OOA(58, 1715, F5, 3, 3) (dual of [(1715, 3), 5137, 4]-NRT-code), using
- appending kth column [i] based on linear OOA(58, 1715, F5, 2, 3) (dual of [(1715, 2), 3422, 4]-NRT-code), using
- linear OOA(551, 2796201, F5, 3, 7) (dual of [(2796201, 3), 8388552, 8]-NRT-code), using
- OOA 3-folding [i] based on linear OA(551, large, F5, 7) (dual of [large, large−51, 8]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 9765624 = 510−1, defining interval I = [0,6], and designed minimum distance d ≥ |I|+1 = 8 [i]
- OOA 3-folding [i] based on linear OA(551, large, F5, 7) (dual of [large, large−51, 8]-code), using
- linear OOA(58, 1715, F5, 3, 3) (dual of [(1715, 3), 5137, 4]-NRT-code), using
- (u, u+v)-construction [i] based on
- OOA stacking with additional row [i] based on linear OOA(559, 2797916, F5, 3, 7) (dual of [(2797916, 3), 8393689, 8]-NRT-code), using
(52, 52+7, large)-Net over F5 — Digital
Digital (52, 59, large)-net over F5, using
- 57 times duplication [i] based on digital (45, 52, large)-net over F5, using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(552, large, F5, 7) (dual of [large, large−52, 8]-code), using
- 1 times code embedding in larger space [i] based on linear OA(551, large, F5, 7) (dual of [large, large−51, 8]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 9765624 = 510−1, defining interval I = [0,6], and designed minimum distance d ≥ |I|+1 = 8 [i]
- 1 times code embedding in larger space [i] based on linear OA(551, large, F5, 7) (dual of [large, large−51, 8]-code), using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(552, large, F5, 7) (dual of [large, large−52, 8]-code), using
(52, 52+7, large)-Net in Base 5 — Upper bound on s
There is no (52, 59, large)-net in base 5, because
- 5 times m-reduction [i] would yield (52, 54, large)-net in base 5, but