Best Known (63−15, 63, s)-Nets in Base 64
(63−15, 63, 1198371)-Net over F64 — Constructive and digital
Digital (48, 63, 1198371)-net over F64, using
- 646 times duplication [i] based on digital (42, 57, 1198371)-net over F64, using
- net defined by OOA [i] based on linear OOA(6457, 1198371, F64, 15, 15) (dual of [(1198371, 15), 17975508, 16]-NRT-code), using
- OOA 7-folding and stacking with additional row [i] based on linear OA(6457, 8388598, F64, 15) (dual of [8388598, 8388541, 16]-code), using
- discarding factors / shortening the dual code 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]
- discarding factors / shortening the dual code based on linear OA(6457, large, F64, 15) (dual of [large, large−57, 16]-code), using
- OOA 7-folding and stacking with additional row [i] based on linear OA(6457, 8388598, F64, 15) (dual of [8388598, 8388541, 16]-code), using
- net defined by OOA [i] based on linear OOA(6457, 1198371, F64, 15, 15) (dual of [(1198371, 15), 17975508, 16]-NRT-code), using
(63−15, 63, large)-Net over F64 — Digital
Digital (48, 63, large)-net over F64, using
- t-expansion [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
(63−15, 63, large)-Net in Base 64 — Upper bound on s
There is no (48, 63, large)-net in base 64, because
- 13 times m-reduction [i] would yield (48, 50, large)-net in base 64, but