Best Known (89−14, 89, s)-Nets in Base 64
(89−14, 89, 2527818)-Net over F64 — Constructive and digital
Digital (75, 89, 2527818)-net over F64, using
- generalized (u, u+v)-construction [i] based on
- digital (7, 11, 131076)-net over F64, using
- net defined by OOA [i] based on linear OOA(6411, 131076, F64, 4, 4) (dual of [(131076, 4), 524293, 5]-NRT-code), using
- appending kth column [i] based on linear OOA(6411, 131076, F64, 3, 4) (dual of [(131076, 3), 393217, 5]-NRT-code), using
- OA 2-folding and stacking [i] based on linear OA(6411, 262152, F64, 4) (dual of [262152, 262141, 5]-code), using
- construction X4 applied to Ce(3) ⊂ Ce(1) [i] based on
- linear OA(6410, 262144, F64, 4) (dual of [262144, 262134, 5]-code), using an extension Ce(3) of the primitive narrow-sense BCH-code C(I) with length 262143 = 643−1, defining interval I = [1,3], and designed minimum distance d ≥ |I|+1 = 4 [i]
- linear OA(644, 262144, F64, 2) (dual of [262144, 262140, 3]-code), using an extension Ce(1) of the primitive narrow-sense BCH-code C(I) with length 262143 = 643−1, defining interval I = [1,1], and designed minimum distance d ≥ |I|+1 = 2 [i]
- linear OA(647, 8, F64, 7) (dual of [8, 1, 8]-code or 8-arc in PG(6,64)), using
- dual of repetition code with length 8 [i]
- linear OA(641, 8, F64, 1) (dual of [8, 7, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(641, 64, F64, 1) (dual of [64, 63, 2]-code), using
- Reed–Solomon code RS(63,64) [i]
- discarding factors / shortening the dual code based on linear OA(641, 64, F64, 1) (dual of [64, 63, 2]-code), using
- construction X4 applied to Ce(3) ⊂ Ce(1) [i] based on
- OA 2-folding and stacking [i] based on linear OA(6411, 262152, F64, 4) (dual of [262152, 262141, 5]-code), using
- appending kth column [i] based on linear OOA(6411, 131076, F64, 3, 4) (dual of [(131076, 3), 393217, 5]-NRT-code), using
- net defined by OOA [i] based on linear OOA(6411, 131076, F64, 4, 4) (dual of [(131076, 4), 524293, 5]-NRT-code), using
- digital (18, 25, 1198371)-net over F64, using
- s-reduction based on digital (18, 25, 2796200)-net over F64, using
- net defined by OOA [i] based on linear OOA(6425, 2796200, F64, 7, 7) (dual of [(2796200, 7), 19573375, 8]-NRT-code), using
- OOA 3-folding and stacking with additional row [i] based on linear OA(6425, 8388601, F64, 7) (dual of [8388601, 8388576, 8]-code), using
- discarding factors / shortening the dual code based on linear OA(6425, large, F64, 7) (dual of [large, large−25, 8]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 16777217 | 648−1, defining interval I = [0,3], and minimum distance d ≥ |{−3,−2,…,3}|+1 = 8 (BCH-bound) [i]
- discarding factors / shortening the dual code based on linear OA(6425, large, F64, 7) (dual of [large, large−25, 8]-code), using
- OOA 3-folding and stacking with additional row [i] based on linear OA(6425, 8388601, F64, 7) (dual of [8388601, 8388576, 8]-code), using
- net defined by OOA [i] based on linear OOA(6425, 2796200, F64, 7, 7) (dual of [(2796200, 7), 19573375, 8]-NRT-code), using
- s-reduction based on digital (18, 25, 2796200)-net over F64, 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
- digital (7, 11, 131076)-net over F64, using
(89−14, 89, large)-Net over F64 — Digital
Digital (75, 89, large)-net over F64, using
- 641 times duplication [i] based on digital (74, 88, large)-net over F64, using
- t-expansion [i] based on digital (66, 88, large)-net over F64, using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(6488, large, F64, 22) (dual of [large, large−88, 23]-code), using
- 3 times code embedding in larger space [i] based on linear OA(6485, large, F64, 22) (dual of [large, large−85, 23]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 16777215 = 644−1, defining interval I = [0,21], and designed minimum distance d ≥ |I|+1 = 23 [i]
- 3 times code embedding in larger space [i] based on linear OA(6485, large, F64, 22) (dual of [large, large−85, 23]-code), using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(6488, large, F64, 22) (dual of [large, large−88, 23]-code), using
- t-expansion [i] based on digital (66, 88, large)-net over F64, using
(89−14, 89, large)-Net in Base 64 — Upper bound on s
There is no (75, 89, large)-net in base 64, because
- 12 times m-reduction [i] would yield (75, 77, large)-net in base 64, but