Best Known (69−12, 69, s)-Nets in Base 64
(69−12, 69, 2796201)-Net over F64 — Constructive and digital
Digital (57, 69, 2796201)-net over F64, using
- net defined by OOA [i] based on linear OOA(6469, 2796201, F64, 15, 12) (dual of [(2796201, 15), 41942946, 13]-NRT-code), using
- OOA 2-folding and stacking with additional row [i] based on linear OOA(6469, 5592403, F64, 3, 12) (dual of [(5592403, 3), 16777140, 13]-NRT-code), using
- 1 times NRT-code embedding in larger space [i] based on linear OOA(6466, 5592402, F64, 3, 12) (dual of [(5592402, 3), 16777140, 13]-NRT-code), using
- (u, u+v)-construction [i] based on
- linear OOA(6421, 2796201, F64, 3, 6) (dual of [(2796201, 3), 8388582, 7]-NRT-code), using
- OOA 3-folding [i] based on linear OA(6421, large, F64, 6) (dual of [large, large−21, 7]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 16777215 = 644−1, defining interval I = [0,5], and designed minimum distance d ≥ |I|+1 = 7 [i]
- OOA 3-folding [i] based on linear OA(6421, large, F64, 6) (dual of [large, large−21, 7]-code), using
- linear OOA(6445, 2796201, F64, 3, 12) (dual of [(2796201, 3), 8388558, 13]-NRT-code), using
- OOA 3-folding [i] based on linear OA(6445, large, F64, 12) (dual of [large, large−45, 13]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 16777215 = 644−1, defining interval I = [0,11], and designed minimum distance d ≥ |I|+1 = 13 [i]
- OOA 3-folding [i] based on linear OA(6445, large, F64, 12) (dual of [large, large−45, 13]-code), using
- linear OOA(6421, 2796201, F64, 3, 6) (dual of [(2796201, 3), 8388582, 7]-NRT-code), using
- (u, u+v)-construction [i] based on
- 1 times NRT-code embedding in larger space [i] based on linear OOA(6466, 5592402, F64, 3, 12) (dual of [(5592402, 3), 16777140, 13]-NRT-code), using
- OOA 2-folding and stacking with additional row [i] based on linear OOA(6469, 5592403, F64, 3, 12) (dual of [(5592403, 3), 16777140, 13]-NRT-code), using
(69−12, 69, large)-Net over F64 — Digital
Digital (57, 69, large)-net over F64, using
- 7 times m-reduction [i] based on digital (57, 76, large)-net over F64, using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(6476, large, F64, 19) (dual of [large, large−76, 20]-code), using
- 3 times code embedding in larger space [i] based on linear OA(6473, large, F64, 19) (dual of [large, large−73, 20]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 16777217 | 648−1, defining interval I = [0,9], and minimum distance d ≥ |{−9,−8,…,9}|+1 = 20 (BCH-bound) [i]
- 3 times code embedding in larger space [i] based on linear OA(6473, large, F64, 19) (dual of [large, large−73, 20]-code), using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(6476, large, F64, 19) (dual of [large, large−76, 20]-code), using
(69−12, 69, large)-Net in Base 64 — Upper bound on s
There is no (57, 69, large)-net in base 64, because
- 10 times m-reduction [i] would yield (57, 59, large)-net in base 64, but