Best Known (104, 114, s)-Nets in Base 8
(104, 114, 6715104)-Net over F8 — Constructive and digital
Digital (104, 114, 6715104)-net over F8, using
- generalized (u, u+v)-construction [i] based on
- digital (4, 7, 4224)-net over F8, using
- net defined by OOA [i] based on linear OOA(87, 4224, F8, 3, 3) (dual of [(4224, 3), 12665, 4]-NRT-code), using
- appending kth column [i] based on linear OOA(87, 4224, F8, 2, 3) (dual of [(4224, 2), 8441, 4]-NRT-code), using
- net defined by OOA [i] based on linear OOA(87, 4224, F8, 3, 3) (dual of [(4224, 3), 12665, 4]-NRT-code), using
- digital (28, 33, 3355440)-net over F8, using
- s-reduction based on digital (28, 33, 4194301)-net over F8, using
- net defined by OOA [i] based on linear OOA(833, 4194301, F8, 5, 5) (dual of [(4194301, 5), 20971472, 6]-NRT-code), using
- OOA 2-folding and stacking with additional row [i] based on linear OA(833, large, F8, 5) (dual of [large, large−33, 6]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 16777217 | 816−1, defining interval I = [0,2], and minimum distance d ≥ |{−2,−1,0,1,2}|+1 = 6 (BCH-bound) [i]
- OOA 2-folding and stacking with additional row [i] based on linear OA(833, large, F8, 5) (dual of [large, large−33, 6]-code), using
- net defined by OOA [i] based on linear OOA(833, 4194301, F8, 5, 5) (dual of [(4194301, 5), 20971472, 6]-NRT-code), using
- s-reduction based on digital (28, 33, 4194301)-net over F8, using
- digital (64, 74, 3355440)-net over F8, using
- trace code for nets [i] based on digital (27, 37, 1677720)-net over F64, using
- net defined by OOA [i] based on linear OOA(6437, 1677720, F64, 10, 10) (dual of [(1677720, 10), 16777163, 11]-NRT-code), using
- OA 5-folding and stacking [i] based on linear OA(6437, 8388600, F64, 10) (dual of [8388600, 8388563, 11]-code), using
- discarding factors / shortening the dual code based on linear OA(6437, large, F64, 10) (dual of [large, large−37, 11]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 16777215 = 644−1, defining interval I = [0,9], and designed minimum distance d ≥ |I|+1 = 11 [i]
- discarding factors / shortening the dual code based on linear OA(6437, large, F64, 10) (dual of [large, large−37, 11]-code), using
- OA 5-folding and stacking [i] based on linear OA(6437, 8388600, F64, 10) (dual of [8388600, 8388563, 11]-code), using
- net defined by OOA [i] based on linear OOA(6437, 1677720, F64, 10, 10) (dual of [(1677720, 10), 16777163, 11]-NRT-code), using
- trace code for nets [i] based on digital (27, 37, 1677720)-net over F64, using
- digital (4, 7, 4224)-net over F8, using
(104, 114, large)-Net over F8 — Digital
Digital (104, 114, large)-net over F8, using
- t-expansion [i] based on digital (100, 114, large)-net over F8, using
- 3 times m-reduction [i] based on digital (100, 117, large)-net over F8, using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(8117, large, F8, 17) (dual of [large, large−117, 18]-code), using
- 4 times code embedding in larger space [i] based on linear OA(8113, large, F8, 17) (dual of [large, large−113, 18]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 16777217 | 816−1, defining interval I = [0,8], and minimum distance d ≥ |{−8,−7,…,8}|+1 = 18 (BCH-bound) [i]
- 4 times code embedding in larger space [i] based on linear OA(8113, large, F8, 17) (dual of [large, large−113, 18]-code), using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(8117, large, F8, 17) (dual of [large, large−117, 18]-code), using
- 3 times m-reduction [i] based on digital (100, 117, large)-net over F8, using
(104, 114, large)-Net in Base 8 — Upper bound on s
There is no (104, 114, large)-net in base 8, because
- 8 times m-reduction [i] would yield (104, 106, large)-net in base 8, but