Best Known (135, 135+10, s)-Nets in Base 5
(135, 135+10, 7148981)-Net over F5 — Constructive and digital
Digital (135, 145, 7148981)-net over F5, using
- generalized (u, u+v)-construction [i] based on
- digital (10, 13, 438101)-net over F5, using
- net defined by OOA [i] based on linear OOA(513, 438101, F5, 3, 3) (dual of [(438101, 3), 1314290, 4]-NRT-code), using
- appending kth column [i] based on linear OOA(513, 438101, F5, 2, 3) (dual of [(438101, 2), 876189, 4]-NRT-code), using
- OAs with strength 3, b ≠ 2, and m > 3 are always embeddable [i] based on linear OA(513, 438101, F5, 3) (dual of [438101, 438088, 4]-code or 438101-cap in PG(12,5)), using
- appending kth column [i] based on linear OOA(513, 438101, F5, 2, 3) (dual of [(438101, 2), 876189, 4]-NRT-code), using
- net defined by OOA [i] based on linear OOA(513, 438101, F5, 3, 3) (dual of [(438101, 3), 1314290, 4]-NRT-code), using
- digital (35, 40, 3355440)-net over F5, using
- s-reduction based on digital (35, 40, 4194301)-net over F5, using
- net defined by OOA [i] based on linear OOA(540, 4194301, F5, 5, 5) (dual of [(4194301, 5), 20971465, 6]-NRT-code), using
- OOA 2-folding and stacking with additional row [i] based on linear OA(540, large, F5, 5) (dual of [large, large−40, 6]-code), using
- the primitive narrow-sense BCH-code C(I) with length 9765624 = 510−1, defining interval I = [1,5], and designed minimum distance d ≥ |I|+1 = 6 [i]
- OOA 2-folding and stacking with additional row [i] based on linear OA(540, large, F5, 5) (dual of [large, large−40, 6]-code), using
- net defined by OOA [i] based on linear OOA(540, 4194301, F5, 5, 5) (dual of [(4194301, 5), 20971465, 6]-NRT-code), using
- s-reduction based on digital (35, 40, 4194301)-net over F5, using
- digital (82, 92, 3355440)-net over F5, using
- trace code for nets [i] based on digital (36, 46, 1677720)-net over F25, using
- net defined by OOA [i] based on linear OOA(2546, 1677720, F25, 10, 10) (dual of [(1677720, 10), 16777154, 11]-NRT-code), using
- OA 5-folding and stacking [i] based on linear OA(2546, 8388600, F25, 10) (dual of [8388600, 8388554, 11]-code), using
- discarding factors / shortening the dual code based on linear OA(2546, large, F25, 10) (dual of [large, large−46, 11]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 9765624 = 255−1, defining interval I = [0,9], and designed minimum distance d ≥ |I|+1 = 11 [i]
- discarding factors / shortening the dual code based on linear OA(2546, large, F25, 10) (dual of [large, large−46, 11]-code), using
- OA 5-folding and stacking [i] based on linear OA(2546, 8388600, F25, 10) (dual of [8388600, 8388554, 11]-code), using
- net defined by OOA [i] based on linear OOA(2546, 1677720, F25, 10, 10) (dual of [(1677720, 10), 16777154, 11]-NRT-code), using
- trace code for nets [i] based on digital (36, 46, 1677720)-net over F25, using
- digital (10, 13, 438101)-net over F5, using
(135, 135+10, large)-Net over F5 — Digital
Digital (135, 145, large)-net over F5, using
- t-expansion [i] based on digital (129, 145, large)-net over F5, using
- 1 times m-reduction [i] based on digital (129, 146, large)-net over F5, using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(5146, large, F5, 17) (dual of [large, large−146, 18]-code), using
- 15 times code embedding in larger space [i] based on linear OA(5131, large, F5, 17) (dual of [large, large−131, 18]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 9765624 = 510−1, defining interval I = [0,16], and designed minimum distance d ≥ |I|+1 = 18 [i]
- 15 times code embedding in larger space [i] based on linear OA(5131, large, F5, 17) (dual of [large, large−131, 18]-code), using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(5146, large, F5, 17) (dual of [large, large−146, 18]-code), using
- 1 times m-reduction [i] based on digital (129, 146, large)-net over F5, using
(135, 135+10, large)-Net in Base 5 — Upper bound on s
There is no (135, 145, large)-net in base 5, because
- 8 times m-reduction [i] would yield (135, 137, large)-net in base 5, but