Best Known (79, 79+12, s)-Nets in Base 64
(79, 79+12, 6990500)-Net over F64 — Constructive and digital
Digital (79, 91, 6990500)-net over F64, using
- generalized (u, u+v)-construction [i] based on
- digital (3, 5, 1398100)-net over F64, using
- s-reduction based on digital (3, 5, large)-net over F64, using
- digital (4, 7, 1398100)-net over F64, using
- s-reduction based on digital (4, 7, large)-net over F64, using
- net defined by OOA [i] based on linear OOA(647, large, F64, 3, 3), using
- appending kth column [i] based on linear OOA(647, large, F64, 2, 3), using
- OAs with strength 3, b ≠ 2, and m > 3 are always embeddable [i] based on linear OA(647, large, F64, 3) (dual of [large, large−7, 4]-code), using
- appending kth column [i] based on linear OOA(647, large, F64, 2, 3), using
- net defined by OOA [i] based on linear OOA(647, large, F64, 3, 3), using
- s-reduction based on digital (4, 7, large)-net over F64, using
- digital (9, 13, 1398100)-net over F64, using
- s-reduction based on digital (9, 13, 4194301)-net over F64, using
- net defined by OOA [i] based on linear OOA(6413, 4194301, F64, 4, 4) (dual of [(4194301, 4), 16777191, 5]-NRT-code), using
- appending kth column [i] 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
- appending kth column [i] based on linear OOA(6413, 4194301, F64, 3, 4) (dual of [(4194301, 3), 12582890, 5]-NRT-code), using
- net defined by OOA [i] based on linear OOA(6413, 4194301, F64, 4, 4) (dual of [(4194301, 4), 16777191, 5]-NRT-code), using
- s-reduction based on digital (9, 13, 4194301)-net over F64, using
- digital (15, 21, 1398100)-net over F64, using
- s-reduction based on digital (15, 21, 2796201)-net over F64, using
- net defined by OOA [i] based on linear OOA(6421, 2796201, F64, 6, 6) (dual of [(2796201, 6), 16777185, 7]-NRT-code), using
- OA 3-folding and stacking [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]
- OA 3-folding and stacking [i] based on linear OA(6421, large, F64, 6) (dual of [large, large−21, 7]-code), using
- net defined by OOA [i] based on linear OOA(6421, 2796201, F64, 6, 6) (dual of [(2796201, 6), 16777185, 7]-NRT-code), using
- s-reduction based on digital (15, 21, 2796201)-net over F64, using
- digital (33, 45, 1398100)-net over F64, using
- net defined by OOA [i] based on linear OOA(6445, 1398100, F64, 12, 12) (dual of [(1398100, 12), 16777155, 13]-NRT-code), using
- OA 6-folding and stacking [i] based on linear OA(6445, 8388600, F64, 12) (dual of [8388600, 8388555, 13]-code), using
- discarding factors / shortening the dual code 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]
- discarding factors / shortening the dual code based on linear OA(6445, large, F64, 12) (dual of [large, large−45, 13]-code), using
- OA 6-folding and stacking [i] based on linear OA(6445, 8388600, F64, 12) (dual of [8388600, 8388555, 13]-code), using
- net defined by OOA [i] based on linear OOA(6445, 1398100, F64, 12, 12) (dual of [(1398100, 12), 16777155, 13]-NRT-code), using
- digital (3, 5, 1398100)-net over F64, using
(79, 79+12, large)-Net over F64 — Digital
Digital (79, 91, large)-net over F64, using
- 643 times duplication [i] based on digital (76, 88, large)-net over F64, using
- t-expansion [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
- t-expansion [i] based on digital (66, 88, large)-net over F64, using
(79, 79+12, large)-Net in Base 64 — Upper bound on s
There is no (79, 91, large)-net in base 64, because
- 10 times m-reduction [i] would yield (79, 81, large)-net in base 64, but