Best Known (76−14, 76, s)-Nets in Base 64
(76−14, 76, 1289850)-Net over F64 — Constructive and digital
Digital (62, 76, 1289850)-net over F64, using
- (u, u+v)-construction [i] based on
- digital (16, 23, 91479)-net over F64, using
- (u, u+v)-construction [i] based on
- digital (1, 4, 4097)-net over F64, using
- net defined by OOA [i] based on linear OOA(644, 4097, F64, 3, 3) (dual of [(4097, 3), 12287, 4]-NRT-code), using
- appending kth column [i] based on linear OOA(644, 4097, F64, 2, 3) (dual of [(4097, 2), 8190, 4]-NRT-code), using
- net defined by OOA [i] based on linear OOA(644, 4097, F64, 3, 3) (dual of [(4097, 3), 12287, 4]-NRT-code), using
- digital (12, 19, 87382)-net over F64, using
- net defined by OOA [i] based on linear OOA(6419, 87382, F64, 7, 7) (dual of [(87382, 7), 611655, 8]-NRT-code), using
- OOA 3-folding and stacking with additional row [i] based on linear OA(6419, 262147, F64, 7) (dual of [262147, 262128, 8]-code), using
- construction X applied to Ce(6) ⊂ Ce(5) [i] based on
- linear OA(6419, 262144, F64, 7) (dual of [262144, 262125, 8]-code), using an extension Ce(6) of the primitive narrow-sense BCH-code C(I) with length 262143 = 643−1, defining interval I = [1,6], and designed minimum distance d ≥ |I|+1 = 7 [i]
- linear OA(6416, 262144, F64, 6) (dual of [262144, 262128, 7]-code), using an extension Ce(5) of the primitive narrow-sense BCH-code C(I) with length 262143 = 643−1, defining interval I = [1,5], and designed minimum distance d ≥ |I|+1 = 6 [i]
- linear OA(640, 3, F64, 0) (dual of [3, 3, 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(6) ⊂ Ce(5) [i] based on
- OOA 3-folding and stacking with additional row [i] based on linear OA(6419, 262147, F64, 7) (dual of [262147, 262128, 8]-code), using
- net defined by OOA [i] based on linear OOA(6419, 87382, F64, 7, 7) (dual of [(87382, 7), 611655, 8]-NRT-code), using
- digital (1, 4, 4097)-net over F64, using
- (u, u+v)-construction [i] based on
- digital (39, 53, 1198371)-net over F64, using
- net defined by OOA [i] based on linear OOA(6453, 1198371, F64, 14, 14) (dual of [(1198371, 14), 16777141, 15]-NRT-code), using
- OA 7-folding and stacking [i] based on linear OA(6453, 8388597, F64, 14) (dual of [8388597, 8388544, 15]-code), using
- discarding factors / shortening the dual code based on linear OA(6453, large, F64, 14) (dual of [large, large−53, 15]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 16777215 = 644−1, defining interval I = [0,13], and designed minimum distance d ≥ |I|+1 = 15 [i]
- discarding factors / shortening the dual code based on linear OA(6453, large, F64, 14) (dual of [large, large−53, 15]-code), using
- OA 7-folding and stacking [i] based on linear OA(6453, 8388597, F64, 14) (dual of [8388597, 8388544, 15]-code), using
- net defined by OOA [i] based on linear OOA(6453, 1198371, F64, 14, 14) (dual of [(1198371, 14), 16777141, 15]-NRT-code), using
- digital (16, 23, 91479)-net over F64, using
(76−14, 76, 1897422)-Net in Base 64 — Constructive
(62, 76, 1897422)-net in base 64, using
- (u, u+v)-construction [i] based on
- (16, 23, 699051)-net in base 64, using
- net defined by OOA [i] based on OOA(6423, 699051, S64, 7, 7), using
- OOA 3-folding and stacking with additional row [i] based on OA(6423, 2097154, S64, 7), using
- discarding factors based on OA(6423, 2097155, S64, 7), using
- discarding parts of the base [i] based on linear OA(12819, 2097155, F128, 7) (dual of [2097155, 2097136, 8]-code), using
- construction X applied to Ce(6) ⊂ Ce(5) [i] based on
- linear OA(12819, 2097152, F128, 7) (dual of [2097152, 2097133, 8]-code), using an extension Ce(6) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 1283−1, defining interval I = [1,6], and designed minimum distance d ≥ |I|+1 = 7 [i]
- linear OA(12816, 2097152, F128, 6) (dual of [2097152, 2097136, 7]-code), using an extension Ce(5) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 1283−1, defining interval I = [1,5], and designed minimum distance d ≥ |I|+1 = 6 [i]
- linear OA(1280, 3, F128, 0) (dual of [3, 3, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(1280, s, F128, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(6) ⊂ Ce(5) [i] based on
- discarding parts of the base [i] based on linear OA(12819, 2097155, F128, 7) (dual of [2097155, 2097136, 8]-code), using
- discarding factors based on OA(6423, 2097155, S64, 7), using
- OOA 3-folding and stacking with additional row [i] based on OA(6423, 2097154, S64, 7), using
- net defined by OOA [i] based on OOA(6423, 699051, S64, 7, 7), using
- digital (39, 53, 1198371)-net over F64, using
- net defined by OOA [i] based on linear OOA(6453, 1198371, F64, 14, 14) (dual of [(1198371, 14), 16777141, 15]-NRT-code), using
- OA 7-folding and stacking [i] based on linear OA(6453, 8388597, F64, 14) (dual of [8388597, 8388544, 15]-code), using
- discarding factors / shortening the dual code based on linear OA(6453, large, F64, 14) (dual of [large, large−53, 15]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 16777215 = 644−1, defining interval I = [0,13], and designed minimum distance d ≥ |I|+1 = 15 [i]
- discarding factors / shortening the dual code based on linear OA(6453, large, F64, 14) (dual of [large, large−53, 15]-code), using
- OA 7-folding and stacking [i] based on linear OA(6453, 8388597, F64, 14) (dual of [8388597, 8388544, 15]-code), using
- net defined by OOA [i] based on linear OOA(6453, 1198371, F64, 14, 14) (dual of [(1198371, 14), 16777141, 15]-NRT-code), using
- (16, 23, 699051)-net in base 64, using
(76−14, 76, large)-Net over F64 — Digital
Digital (62, 76, large)-net over F64, using
- t-expansion [i] based on digital (60, 76, large)-net over F64, using
- 4 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
- 4 times m-reduction [i] based on digital (60, 80, large)-net over F64, using
(76−14, 76, large)-Net in Base 64 — Upper bound on s
There is no (62, 76, large)-net in base 64, because
- 12 times m-reduction [i] would yield (62, 64, large)-net in base 64, but