Best Known (133, 133+10, s)-Nets in Base 5
(133, 133+10, 6754756)-Net over F5 — Constructive and digital
Digital (133, 143, 6754756)-net over F5, using
- generalized (u, u+v)-construction [i] based on
- digital (8, 11, 43876)-net over F5, using
- net defined by OOA [i] based on linear OOA(511, 43876, F5, 3, 3) (dual of [(43876, 3), 131617, 4]-NRT-code), using
- appending kth column [i] based on linear OOA(511, 43876, F5, 2, 3) (dual of [(43876, 2), 87741, 4]-NRT-code), using
- OAs with strength 3, b ≠ 2, and m > 3 are always embeddable [i] based on linear OA(511, 43876, F5, 3) (dual of [43876, 43865, 4]-code or 43876-cap in PG(10,5)), using
- appending kth column [i] based on linear OOA(511, 43876, F5, 2, 3) (dual of [(43876, 2), 87741, 4]-NRT-code), using
- net defined by OOA [i] based on linear OOA(511, 43876, F5, 3, 3) (dual of [(43876, 3), 131617, 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 (8, 11, 43876)-net over F5, using
(133, 133+10, large)-Net over F5 — Digital
Digital (133, 143, large)-net over F5, using
- t-expansion [i] based on digital (129, 143, large)-net over F5, using
- 3 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
- 3 times m-reduction [i] based on digital (129, 146, large)-net over F5, using
(133, 133+10, large)-Net in Base 5 — Upper bound on s
There is no (133, 143, large)-net in base 5, because
- 8 times m-reduction [i] would yield (133, 135, large)-net in base 5, but