Best Known (49, 61, s)-Nets in Base 64
(49, 61, 1485482)-Net over F64 — Constructive and digital
Digital (49, 61, 1485482)-net over F64, using
- (u, u+v)-construction [i] based on
- digital (10, 16, 87382)-net over F64, using
- net defined by OOA [i] based on linear OOA(6416, 87382, F64, 6, 6) (dual of [(87382, 6), 524276, 7]-NRT-code), using
- OA 3-folding and stacking [i] based on linear OA(6416, 262146, F64, 6) (dual of [262146, 262130, 7]-code), using
- discarding factors / shortening the dual code based on linear OA(6416, 262147, F64, 6) (dual of [262147, 262131, 7]-code), using
- construction X applied to Ce(5) ⊂ Ce(4) [i] based on
- 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(6413, 262144, F64, 5) (dual of [262144, 262131, 6]-code), using an extension Ce(4) of the primitive narrow-sense BCH-code C(I) with length 262143 = 643−1, defining interval I = [1,4], and designed minimum distance d ≥ |I|+1 = 5 [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(5) ⊂ Ce(4) [i] based on
- discarding factors / shortening the dual code based on linear OA(6416, 262147, F64, 6) (dual of [262147, 262131, 7]-code), using
- OA 3-folding and stacking [i] based on linear OA(6416, 262146, F64, 6) (dual of [262146, 262130, 7]-code), using
- net defined by OOA [i] based on linear OOA(6416, 87382, F64, 6, 6) (dual of [(87382, 6), 524276, 7]-NRT-code), 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 (10, 16, 87382)-net over F64, using
(49, 61, large)-Net over F64 — Digital
Digital (49, 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
(49, 61, large)-Net in Base 64 — Upper bound on s
There is no (49, 61, large)-net in base 64, because
- 10 times m-reduction [i] would yield (49, 51, large)-net in base 64, but