Best Known (137, 137+15, s)-Nets in Base 8
(137, 137+15, 2571506)-Net over F8 — Constructive and digital
Digital (137, 152, 2571506)-net over F8, using
- (u, u+v)-construction [i] based on
- digital (31, 38, 174764)-net over F8, using
- net defined by OOA [i] based on linear OOA(838, 174764, F8, 7, 7) (dual of [(174764, 7), 1223310, 8]-NRT-code), using
- OOA 3-folding and stacking with additional row [i] based on linear OA(838, 524293, F8, 7) (dual of [524293, 524255, 8]-code), using
- discarding factors / shortening the dual code based on linear OA(838, 524294, F8, 7) (dual of [524294, 524256, 8]-code), using
- trace code [i] based on linear OA(6419, 262147, F64, 7) (dual of [262147, 262128, 8]-code), using
- construction X applied to Ce(6) ⊂ Ce(5) [i] based on
- linear OA(6419, 262144, F64, 7) (dual of [262144, 262125, 8]-code), using an extension Ce(6) of the primitive narrow-sense BCH-code C(I) with length 262143 = 643−1, defining interval I = [1,6], and designed minimum distance d ≥ |I|+1 = 7 [i]
- linear OA(6416, 262144, F64, 6) (dual of [262144, 262128, 7]-code), using an extension Ce(5) of the primitive narrow-sense BCH-code C(I) with length 262143 = 643−1, defining interval I = [1,5], and designed minimum distance d ≥ |I|+1 = 6 [i]
- linear OA(640, 3, F64, 0) (dual of [3, 3, 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(6419, 262147, F64, 7) (dual of [262147, 262128, 8]-code), using
- discarding factors / shortening the dual code based on linear OA(838, 524294, F8, 7) (dual of [524294, 524256, 8]-code), using
- OOA 3-folding and stacking with additional row [i] based on linear OA(838, 524293, F8, 7) (dual of [524293, 524255, 8]-code), using
- net defined by OOA [i] based on linear OOA(838, 174764, F8, 7, 7) (dual of [(174764, 7), 1223310, 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 (31, 38, 174764)-net over F8, using
(137, 137+15, large)-Net over F8 — Digital
Digital (137, 152, large)-net over F8, using
- t-expansion [i] based on digital (131, 152, large)-net over F8, using
- 1 times m-reduction [i] based on digital (131, 153, large)-net over F8, using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(8153, large, F8, 22) (dual of [large, large−153, 23]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 16777215 = 88−1, defining interval I = [0,21], and designed minimum distance d ≥ |I|+1 = 23 [i]
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(8153, large, F8, 22) (dual of [large, large−153, 23]-code), using
- 1 times m-reduction [i] based on digital (131, 153, large)-net over F8, using
(137, 137+15, large)-Net in Base 8 — Upper bound on s
There is no (137, 152, large)-net in base 8, because
- 13 times m-reduction [i] would yield (137, 139, large)-net in base 8, but