Best Known (57, 57+7, s)-Nets in Base 4
(57, 57+7, 2796205)-Net over F4 — Constructive and digital
Digital (57, 64, 2796205)-net over F4, using
- net defined by OOA [i] based on linear OOA(464, 2796205, F4, 9, 7) (dual of [(2796205, 9), 25165781, 8]-NRT-code), using
- OOA stacking with additional row [i] based on linear OOA(464, 2796206, F4, 3, 7) (dual of [(2796206, 3), 8388554, 8]-NRT-code), using
- (u, u+v)-construction [i] based on
- linear OOA(43, 5, F4, 3, 3) (dual of [(5, 3), 12, 4]-NRT-code), using
- extended Reed–Solomon NRT-code RSe(3;12,4) [i]
- linear OOA(461, 2796201, F4, 3, 7) (dual of [(2796201, 3), 8388542, 8]-NRT-code), using
- OOA 3-folding [i] based on linear OA(461, large, F4, 7) (dual of [large, large−61, 8]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 16777215 = 412−1, defining interval I = [0,6], and designed minimum distance d ≥ |I|+1 = 8 [i]
- OOA 3-folding [i] based on linear OA(461, large, F4, 7) (dual of [large, large−61, 8]-code), using
- linear OOA(43, 5, F4, 3, 3) (dual of [(5, 3), 12, 4]-NRT-code), using
- (u, u+v)-construction [i] based on
- OOA stacking with additional row [i] based on linear OOA(464, 2796206, F4, 3, 7) (dual of [(2796206, 3), 8388554, 8]-NRT-code), using
(57, 57+7, large)-Net over F4 — Digital
Digital (57, 64, large)-net over F4, using
- 43 times duplication [i] based on digital (54, 61, large)-net over F4, using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(461, large, F4, 7) (dual of [large, large−61, 8]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 16777215 = 412−1, defining interval I = [0,6], and designed minimum distance d ≥ |I|+1 = 8 [i]
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(461, large, F4, 7) (dual of [large, large−61, 8]-code), using
(57, 57+7, large)-Net in Base 4 — Upper bound on s
There is no (57, 64, large)-net in base 4, because
- 5 times m-reduction [i] would yield (57, 59, large)-net in base 4, but