Best Known (48, 48+13, s)-Nets in Base 64
(48, 48+13, 1399467)-Net over F64 — Constructive and digital
Digital (48, 61, 1399467)-net over F64, using
- (u, u+v)-construction [i] based on
- digital (6, 12, 1367)-net over F64, using
- net defined by OOA [i] based on linear OOA(6412, 1367, F64, 6, 6) (dual of [(1367, 6), 8190, 7]-NRT-code), using
- OA 3-folding and stacking [i] based on linear OA(6412, 4101, F64, 6) (dual of [4101, 4089, 7]-code), using
- construction X applied to Ce(5) ⊂ Ce(3) [i] based on
- 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(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(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(5) ⊂ Ce(3) [i] based on
- OA 3-folding and stacking [i] based on linear OA(6412, 4101, F64, 6) (dual of [4101, 4089, 7]-code), using
- net defined by OOA [i] based on linear OOA(6412, 1367, F64, 6, 6) (dual of [(1367, 6), 8190, 7]-NRT-code), using
- digital (36, 49, 1398100)-net over F64, using
- net defined by OOA [i] based on linear OOA(6449, 1398100, F64, 13, 13) (dual of [(1398100, 13), 18175251, 14]-NRT-code), using
- OOA 6-folding and stacking with additional row [i] based on linear OA(6449, 8388601, F64, 13) (dual of [8388601, 8388552, 14]-code), using
- discarding factors / shortening the dual code 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]
- discarding factors / shortening the dual code based on linear OA(6449, large, F64, 13) (dual of [large, large−49, 14]-code), using
- OOA 6-folding and stacking with additional row [i] based on linear OA(6449, 8388601, F64, 13) (dual of [8388601, 8388552, 14]-code), using
- net defined by OOA [i] based on linear OOA(6449, 1398100, F64, 13, 13) (dual of [(1398100, 13), 18175251, 14]-NRT-code), using
- digital (6, 12, 1367)-net over F64, using
(48, 48+13, large)-Net over F64 — Digital
Digital (48, 61, large)-net over F64, using
- t-expansion [i] based on digital (47, 61, large)-net over F64, using
- 2 times m-reduction [i] based on digital (47, 63, large)-net over F64, using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(6463, large, F64, 16) (dual of [large, large−63, 17]-code), using
- 2 times code embedding in larger space [i] 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]
- 2 times code embedding in larger space [i] based on linear OA(6461, large, F64, 16) (dual of [large, large−61, 17]-code), using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(6463, large, F64, 16) (dual of [large, large−63, 17]-code), using
- 2 times m-reduction [i] based on digital (47, 63, large)-net over F64, using
(48, 48+13, large)-Net in Base 64 — Upper bound on s
There is no (48, 61, large)-net in base 64, because
- 11 times m-reduction [i] would yield (48, 50, large)-net in base 64, but