Best Known (77−16, 77, s)-Nets in Base 64
(77−16, 77, 1049600)-Net over F64 — Constructive and digital
Digital (61, 77, 1049600)-net over F64, using
- (u, u+v)-construction [i] based on
- digital (8, 16, 1025)-net over F64, using
- net defined by OOA [i] based on linear OOA(6416, 1025, F64, 8, 8) (dual of [(1025, 8), 8184, 9]-NRT-code), using
- OA 4-folding and stacking [i] based on linear OA(6416, 4100, F64, 8) (dual of [4100, 4084, 9]-code), using
- discarding factors / shortening the dual code based on linear OA(6416, 4101, F64, 8) (dual of [4101, 4085, 9]-code), using
- construction X applied to Ce(7) ⊂ Ce(5) [i] based on
- linear OA(6415, 4096, F64, 8) (dual of [4096, 4081, 9]-code), using an extension Ce(7) of the primitive narrow-sense BCH-code C(I) with length 4095 = 642−1, defining interval I = [1,7], and designed minimum distance d ≥ |I|+1 = 8 [i]
- linear OA(6411, 4096, F64, 6) (dual of [4096, 4085, 7]-code), using an extension Ce(5) of the primitive narrow-sense BCH-code C(I) with length 4095 = 642−1, defining interval I = [1,5], and designed minimum distance d ≥ |I|+1 = 6 [i]
- linear OA(641, 5, F64, 1) (dual of [5, 4, 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
- construction X applied to Ce(7) ⊂ Ce(5) [i] based on
- discarding factors / shortening the dual code based on linear OA(6416, 4101, F64, 8) (dual of [4101, 4085, 9]-code), using
- OA 4-folding and stacking [i] based on linear OA(6416, 4100, F64, 8) (dual of [4100, 4084, 9]-code), using
- net defined by OOA [i] based on linear OOA(6416, 1025, F64, 8, 8) (dual of [(1025, 8), 8184, 9]-NRT-code), using
- digital (45, 61, 1048575)-net over F64, using
- net defined by OOA [i] based on linear OOA(6461, 1048575, F64, 16, 16) (dual of [(1048575, 16), 16777139, 17]-NRT-code), using
- OA 8-folding and stacking [i] based on linear OA(6461, 8388600, F64, 16) (dual of [8388600, 8388539, 17]-code), using
- discarding factors / shortening the dual code based on linear OA(6461, large, F64, 16) (dual of [large, large−61, 17]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 16777215 = 644−1, defining interval I = [0,15], and designed minimum distance d ≥ |I|+1 = 17 [i]
- discarding factors / shortening the dual code based on linear OA(6461, large, F64, 16) (dual of [large, large−61, 17]-code), using
- OA 8-folding and stacking [i] based on linear OA(6461, 8388600, F64, 16) (dual of [8388600, 8388539, 17]-code), using
- net defined by OOA [i] based on linear OOA(6461, 1048575, F64, 16, 16) (dual of [(1048575, 16), 16777139, 17]-NRT-code), using
- digital (8, 16, 1025)-net over F64, using
(77−16, 77, large)-Net over F64 — Digital
Digital (61, 77, large)-net over F64, using
- t-expansion [i] based on digital (60, 77, large)-net over F64, using
- 3 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
- 3 times m-reduction [i] based on digital (60, 80, large)-net over F64, using
(77−16, 77, large)-Net in Base 64 — Upper bound on s
There is no (61, 77, large)-net in base 64, because
- 14 times m-reduction [i] would yield (61, 63, large)-net in base 64, but