Best Known (126, 126+15, s)-Nets in Base 8
(126, 126+15, 2399474)-Net over F8 — Constructive and digital
Digital (126, 141, 2399474)-net over F8, using
- (u, u+v)-construction [i] based on
- digital (20, 27, 2732)-net over F8, using
- net defined by OOA [i] based on linear OOA(827, 2732, F8, 7, 7) (dual of [(2732, 7), 19097, 8]-NRT-code), using
- OOA 3-folding and stacking with additional row [i] based on linear OA(827, 8197, F8, 7) (dual of [8197, 8170, 8]-code), using
- 1 times code embedding in larger space [i] based on linear OA(826, 8196, F8, 7) (dual of [8196, 8170, 8]-code), using
- trace code [i] based on linear OA(6413, 4098, F64, 7) (dual of [4098, 4085, 8]-code), using
- construction X applied to Ce(6) ⊂ Ce(5) [i] based on
- linear OA(6413, 4096, F64, 7) (dual of [4096, 4083, 8]-code), using an extension Ce(6) of the primitive narrow-sense BCH-code C(I) with length 4095 = 642−1, defining interval I = [1,6], and designed minimum distance d ≥ |I|+1 = 7 [i]
- 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(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(6) ⊂ Ce(5) [i] based on
- trace code [i] based on linear OA(6413, 4098, F64, 7) (dual of [4098, 4085, 8]-code), using
- 1 times code embedding in larger space [i] based on linear OA(826, 8196, F8, 7) (dual of [8196, 8170, 8]-code), using
- OOA 3-folding and stacking with additional row [i] based on linear OA(827, 8197, F8, 7) (dual of [8197, 8170, 8]-code), using
- net defined by OOA [i] based on linear OOA(827, 2732, F8, 7, 7) (dual of [(2732, 7), 19097, 8]-NRT-code), using
- digital (99, 114, 2396742)-net over F8, using
- trace code for nets [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
- trace code for nets [i] based on digital (42, 57, 1198371)-net over F64, using
- digital (20, 27, 2732)-net over F8, using
(126, 126+15, large)-Net over F8 — Digital
Digital (126, 141, large)-net over F8, using
- t-expansion [i] based on digital (125, 141, large)-net over F8, using
- 5 times m-reduction [i] based on digital (125, 146, large)-net over F8, using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(8146, large, F8, 21) (dual of [large, large−146, 22]-code), using
- 1 times code embedding in larger space [i] based on linear OA(8145, large, F8, 21) (dual of [large, large−145, 22]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 16777217 | 816−1, defining interval I = [0,10], and minimum distance d ≥ |{−10,−9,…,10}|+1 = 22 (BCH-bound) [i]
- 1 times code embedding in larger space [i] based on linear OA(8145, large, F8, 21) (dual of [large, large−145, 22]-code), using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(8146, large, F8, 21) (dual of [large, large−146, 22]-code), using
- 5 times m-reduction [i] based on digital (125, 146, large)-net over F8, using
(126, 126+15, large)-Net in Base 8 — Upper bound on s
There is no (126, 141, large)-net in base 8, because
- 13 times m-reduction [i] would yield (126, 128, large)-net in base 8, but