Best Known (85−13, 85, s)-Nets in Base 64
(85−13, 85, 4194301)-Net over F64 — Constructive and digital
Digital (72, 85, 4194301)-net over F64, using
- 642 times duplication [i] based on digital (70, 83, 4194301)-net over F64, using
- net defined by OOA [i] based on linear OOA(6483, 4194301, F64, 15, 13) (dual of [(4194301, 15), 62914432, 14]-NRT-code), using
- OOA 2-folding and stacking with additional row [i] based on linear OOA(6483, large, F64, 3, 13), using
- generalized (u, u+v)-construction [i] based on
- linear OOA(6413, 2796201, F64, 3, 4) (dual of [(2796201, 3), 8388590, 5]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(6413, 4194301, F64, 3, 4) (dual of [(4194301, 3), 12582890, 5]-NRT-code), using
- OA 2-folding and stacking [i] based on linear OA(6413, 8388602, F64, 4) (dual of [8388602, 8388589, 5]-code), using
- discarding factors / shortening the dual code based on linear OA(6413, large, F64, 4) (dual of [large, large−13, 5]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 16777215 = 644−1, defining interval I = [0,3], and designed minimum distance d ≥ |I|+1 = 5 [i]
- discarding factors / shortening the dual code based on linear OA(6413, large, F64, 4) (dual of [large, large−13, 5]-code), using
- OA 2-folding and stacking [i] based on linear OA(6413, 8388602, F64, 4) (dual of [8388602, 8388589, 5]-code), using
- discarding factors / shortening the dual code based on linear OOA(6413, 4194301, F64, 3, 4) (dual of [(4194301, 3), 12582890, 5]-NRT-code), using
- 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(6449, 2796201, F64, 3, 13) (dual of [(2796201, 3), 8388554, 14]-NRT-code), using
- OOA 3-folding [i] based on linear OA(6449, large, F64, 13) (dual of [large, large−49, 14]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 16777217 | 648−1, defining interval I = [0,6], and minimum distance d ≥ |{−6,−5,…,6}|+1 = 14 (BCH-bound) [i]
- OOA 3-folding [i] based on linear OA(6449, large, F64, 13) (dual of [large, large−49, 14]-code), using
- linear OOA(6413, 2796201, F64, 3, 4) (dual of [(2796201, 3), 8388590, 5]-NRT-code), using
- generalized (u, u+v)-construction [i] based on
- OOA 2-folding and stacking with additional row [i] based on linear OOA(6483, large, F64, 3, 13), using
- net defined by OOA [i] based on linear OOA(6483, 4194301, F64, 15, 13) (dual of [(4194301, 15), 62914432, 14]-NRT-code), using
(85−13, 85, large)-Net over F64 — Digital
Digital (72, 85, large)-net over F64, using
- t-expansion [i] based on digital (66, 85, large)-net over F64, using
- 3 times m-reduction [i] based on digital (66, 88, large)-net over F64, using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(6488, large, F64, 22) (dual of [large, large−88, 23]-code), using
- 3 times code embedding in larger space [i] based on linear OA(6485, large, F64, 22) (dual of [large, large−85, 23]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 16777215 = 644−1, defining interval I = [0,21], and designed minimum distance d ≥ |I|+1 = 23 [i]
- 3 times code embedding in larger space [i] based on linear OA(6485, large, F64, 22) (dual of [large, large−85, 23]-code), using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(6488, large, F64, 22) (dual of [large, large−88, 23]-code), using
- 3 times m-reduction [i] based on digital (66, 88, large)-net over F64, using
(85−13, 85, large)-Net in Base 64 — Upper bound on s
There is no (72, 85, large)-net in base 64, because
- 11 times m-reduction [i] would yield (72, 74, large)-net in base 64, but