Best Known (39, 39+10, s)-Nets in Base 64
(39, 39+10, 1685785)-Net over F64 — Constructive and digital
Digital (39, 49, 1685785)-net over F64, using
- (u, u+v)-construction [i] based on
- digital (7, 12, 8065)-net over F64, using
- net defined by OOA [i] based on linear OOA(6412, 8065, F64, 5, 5) (dual of [(8065, 5), 40313, 6]-NRT-code), using
- OOA 2-folding and stacking with additional row [i] based on linear OA(6412, 16131, F64, 5) (dual of [16131, 16119, 6]-code), using
- discarding factors / shortening the dual code based on linear OA(6412, 16132, F64, 5) (dual of [16132, 16120, 6]-code), using
- generalized (u, u+v)-construction [i] based on
- linear OA(641, 4033, F64, 1) (dual of [4033, 4032, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(641, s, F64, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- linear OA(641, 4033, F64, 1) (dual of [4033, 4032, 2]-code) (see above)
- linear OA(643, 4033, F64, 2) (dual of [4033, 4030, 3]-code), using
- discarding factors / shortening the dual code based on linear OA(643, 4161, F64, 2) (dual of [4161, 4158, 3]-code), using
- Hamming code H(3,64) [i]
- discarding factors / shortening the dual code based on linear OA(643, 4161, F64, 2) (dual of [4161, 4158, 3]-code), using
- linear OA(647, 4033, F64, 5) (dual of [4033, 4026, 6]-code), using
- linear OA(641, 4033, F64, 1) (dual of [4033, 4032, 2]-code), using
- generalized (u, u+v)-construction [i] based on
- discarding factors / shortening the dual code based on linear OA(6412, 16132, F64, 5) (dual of [16132, 16120, 6]-code), using
- OOA 2-folding and stacking with additional row [i] based on linear OA(6412, 16131, F64, 5) (dual of [16131, 16119, 6]-code), using
- net defined by OOA [i] based on linear OOA(6412, 8065, F64, 5, 5) (dual of [(8065, 5), 40313, 6]-NRT-code), using
- digital (27, 37, 1677720)-net over F64, using
- net defined by OOA [i] based on linear OOA(6437, 1677720, F64, 10, 10) (dual of [(1677720, 10), 16777163, 11]-NRT-code), using
- OA 5-folding and stacking [i] based on linear OA(6437, 8388600, F64, 10) (dual of [8388600, 8388563, 11]-code), using
- discarding factors / shortening the dual code based on linear OA(6437, large, F64, 10) (dual of [large, large−37, 11]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 16777215 = 644−1, defining interval I = [0,9], and designed minimum distance d ≥ |I|+1 = 11 [i]
- discarding factors / shortening the dual code based on linear OA(6437, large, F64, 10) (dual of [large, large−37, 11]-code), using
- OA 5-folding and stacking [i] based on linear OA(6437, 8388600, F64, 10) (dual of [8388600, 8388563, 11]-code), using
- net defined by OOA [i] based on linear OOA(6437, 1677720, F64, 10, 10) (dual of [(1677720, 10), 16777163, 11]-NRT-code), using
- digital (7, 12, 8065)-net over F64, using
(39, 39+10, 1710617)-Net in Base 64 — Constructive
(39, 49, 1710617)-net in base 64, using
- (u, u+v)-construction [i] based on
- (7, 12, 32897)-net in base 64, using
- base change [i] based on digital (4, 9, 32897)-net over F256, using
- net defined by OOA [i] based on linear OOA(2569, 32897, F256, 5, 5) (dual of [(32897, 5), 164476, 6]-NRT-code), using
- appending kth column [i] based on linear OOA(2569, 32897, F256, 4, 5) (dual of [(32897, 4), 131579, 6]-NRT-code), using
- (u, u+v)-construction [i] based on
- linear OOA(2562, 257, F256, 4, 2) (dual of [(257, 4), 1026, 3]-NRT-code), using
- extended Reed–Solomon NRT-code RSe(4;1026,256) [i]
- linear OOA(2567, 32640, F256, 4, 5) (dual of [(32640, 4), 130553, 6]-NRT-code), using
- OOA 2-folding and stacking with additional row [i] based on linear OA(2567, 65281, F256, 5) (dual of [65281, 65274, 6]-code), using
- linear OOA(2562, 257, F256, 4, 2) (dual of [(257, 4), 1026, 3]-NRT-code), using
- (u, u+v)-construction [i] based on
- appending kth column [i] based on linear OOA(2569, 32897, F256, 4, 5) (dual of [(32897, 4), 131579, 6]-NRT-code), using
- net defined by OOA [i] based on linear OOA(2569, 32897, F256, 5, 5) (dual of [(32897, 5), 164476, 6]-NRT-code), using
- base change [i] based on digital (4, 9, 32897)-net over F256, using
- digital (27, 37, 1677720)-net over F64, using
- net defined by OOA [i] based on linear OOA(6437, 1677720, F64, 10, 10) (dual of [(1677720, 10), 16777163, 11]-NRT-code), using
- OA 5-folding and stacking [i] based on linear OA(6437, 8388600, F64, 10) (dual of [8388600, 8388563, 11]-code), using
- discarding factors / shortening the dual code based on linear OA(6437, large, F64, 10) (dual of [large, large−37, 11]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 16777215 = 644−1, defining interval I = [0,9], and designed minimum distance d ≥ |I|+1 = 11 [i]
- discarding factors / shortening the dual code based on linear OA(6437, large, F64, 10) (dual of [large, large−37, 11]-code), using
- OA 5-folding and stacking [i] based on linear OA(6437, 8388600, F64, 10) (dual of [8388600, 8388563, 11]-code), using
- net defined by OOA [i] based on linear OOA(6437, 1677720, F64, 10, 10) (dual of [(1677720, 10), 16777163, 11]-NRT-code), using
- (7, 12, 32897)-net in base 64, using
(39, 39+10, large)-Net over F64 — Digital
Digital (39, 49, large)-net over F64, using
- t-expansion [i] based on digital (37, 49, large)-net over F64, using
- 1 times m-reduction [i] based on digital (37, 50, large)-net over F64, using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(6450, large, F64, 13) (dual of [large, large−50, 14]-code), using
- 1 times code embedding in larger space [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]
- 1 times code embedding in larger space [i] based on linear OA(6449, large, F64, 13) (dual of [large, large−49, 14]-code), using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(6450, large, F64, 13) (dual of [large, large−50, 14]-code), using
- 1 times m-reduction [i] based on digital (37, 50, large)-net over F64, using
(39, 39+10, large)-Net in Base 64 — Upper bound on s
There is no (39, 49, large)-net in base 64, because
- 8 times m-reduction [i] would yield (39, 41, large)-net in base 64, but