Best Known (61, 61+12, s)-Nets in Base 64
(61, 61+12, 2798249)-Net over F64 — Constructive and digital
Digital (61, 73, 2798249)-net over F64, using
- generalized (u, u+v)-construction [i] based on
- digital (3, 7, 2049)-net over F64, using
- net defined by OOA [i] based on linear OOA(647, 2049, F64, 4, 4) (dual of [(2049, 4), 8189, 5]-NRT-code), using
- appending kth column [i] based on linear OOA(647, 2049, F64, 3, 4) (dual of [(2049, 3), 6140, 5]-NRT-code), using
- OA 2-folding and stacking [i] based on linear OA(647, 4098, F64, 4) (dual of [4098, 4091, 5]-code), using
- construction X applied to Ce(3) ⊂ Ce(2) [i] based on
- linear OA(647, 4096, F64, 4) (dual of [4096, 4089, 5]-code), using an extension Ce(3) of the primitive narrow-sense BCH-code C(I) with length 4095 = 642−1, defining interval I = [1,3], and designed minimum distance d ≥ |I|+1 = 4 [i]
- linear OA(645, 4096, F64, 3) (dual of [4096, 4091, 4]-code or 4096-cap in PG(4,64)), using an extension Ce(2) of the primitive narrow-sense BCH-code C(I) with length 4095 = 642−1, defining interval I = [1,2], and designed minimum distance d ≥ |I|+1 = 3 [i]
- linear OA(640, 2, F64, 0) (dual of [2, 2, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(640, s, F64, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(3) ⊂ Ce(2) [i] based on
- OA 2-folding and stacking [i] based on linear OA(647, 4098, F64, 4) (dual of [4098, 4091, 5]-code), using
- appending kth column [i] based on linear OOA(647, 2049, F64, 3, 4) (dual of [(2049, 3), 6140, 5]-NRT-code), using
- net defined by OOA [i] based on linear OOA(647, 2049, F64, 4, 4) (dual of [(2049, 4), 8189, 5]-NRT-code), 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, 7, 2049)-net over F64, using
(61, 61+12, large)-Net over F64 — Digital
Digital (61, 73, large)-net over F64, using
- t-expansion [i] based on digital (60, 73, large)-net over F64, using
- 7 times m-reduction [i] based on digital (60, 80, large)-net over F64, using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(6480, large, F64, 20) (dual of [large, large−80, 21]-code), using
- 3 times code embedding in larger space [i] based on linear OA(6477, large, F64, 20) (dual of [large, large−77, 21]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 16777215 = 644−1, defining interval I = [0,19], and designed minimum distance d ≥ |I|+1 = 21 [i]
- 3 times code embedding in larger space [i] based on linear OA(6477, large, F64, 20) (dual of [large, large−77, 21]-code), using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(6480, large, F64, 20) (dual of [large, large−80, 21]-code), using
- 7 times m-reduction [i] based on digital (60, 80, large)-net over F64, using
(61, 61+12, large)-Net in Base 64 — Upper bound on s
There is no (61, 73, large)-net in base 64, because
- 10 times m-reduction [i] would yield (61, 63, large)-net in base 64, but