Best Known (44, 56, s)-Nets in Base 64
(44, 56, 1399466)-Net over F64 — Constructive and digital
Digital (44, 56, 1399466)-net over F64, using
- (u, u+v)-construction [i] based on
- digital (5, 11, 1366)-net over F64, using
- net defined by OOA [i] based on linear OOA(6411, 1366, F64, 6, 6) (dual of [(1366, 6), 8185, 7]-NRT-code), using
- OA 3-folding and stacking [i] based on linear OA(6411, 4098, F64, 6) (dual of [4098, 4087, 7]-code), using
- construction X applied to Ce(5) ⊂ Ce(4) [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(649, 4096, F64, 5) (dual of [4096, 4087, 6]-code), using an extension Ce(4) of the primitive narrow-sense BCH-code C(I) with length 4095 = 642−1, defining interval I = [1,4], and designed minimum distance d ≥ |I|+1 = 5 [i]
- linear OA(640, 2, F64, 0) (dual of [2, 2, 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
- OA 3-folding and stacking [i] based on linear OA(6411, 4098, F64, 6) (dual of [4098, 4087, 7]-code), using
- net defined by OOA [i] based on linear OOA(6411, 1366, F64, 6, 6) (dual of [(1366, 6), 8185, 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 (5, 11, 1366)-net over F64, using
(44, 56, large)-Net over F64 — Digital
Digital (44, 56, large)-net over F64, using
- 3 times m-reduction [i] based on digital (44, 59, large)-net over F64, using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(6459, large, F64, 15) (dual of [large, large−59, 16]-code), using
- 2 times code embedding in larger space [i] based on linear OA(6457, large, F64, 15) (dual of [large, large−57, 16]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 16777217 | 648−1, defining interval I = [0,7], and minimum distance d ≥ |{−7,−6,…,7}|+1 = 16 (BCH-bound) [i]
- 2 times code embedding in larger space [i] based on linear OA(6457, large, F64, 15) (dual of [large, large−57, 16]-code), using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(6459, large, F64, 15) (dual of [large, large−59, 16]-code), using
(44, 56, large)-Net in Base 64 — Upper bound on s
There is no (44, 56, large)-net in base 64, because
- 10 times m-reduction [i] would yield (44, 46, large)-net in base 64, but