Best Known (134, 134+13, s)-Nets in Base 5
(134, 134+13, 2801408)-Net over F5 — Constructive and digital
Digital (134, 147, 2801408)-net over F5, using
- (u, u+v)-construction [i] based on
- digital (19, 25, 5208)-net over F5, using
- net defined by OOA [i] based on linear OOA(525, 5208, F5, 6, 6) (dual of [(5208, 6), 31223, 7]-NRT-code), using
- OA 3-folding and stacking [i] based on linear OA(525, 15624, F5, 6) (dual of [15624, 15599, 7]-code), using
- discarding factors / shortening the dual code based on linear OA(525, 15625, F5, 6) (dual of [15625, 15600, 7]-code), using
- an extension Ce(5) of the primitive narrow-sense BCH-code C(I) with length 15624 = 56−1, defining interval I = [1,5], and designed minimum distance d ≥ |I|+1 = 6 [i]
- discarding factors / shortening the dual code based on linear OA(525, 15625, F5, 6) (dual of [15625, 15600, 7]-code), using
- OA 3-folding and stacking [i] based on linear OA(525, 15624, F5, 6) (dual of [15624, 15599, 7]-code), using
- net defined by OOA [i] based on linear OOA(525, 5208, F5, 6, 6) (dual of [(5208, 6), 31223, 7]-NRT-code), using
- digital (109, 122, 2796200)-net over F5, using
- net defined by OOA [i] based on linear OOA(5122, 2796200, F5, 14, 13) (dual of [(2796200, 14), 39146678, 14]-NRT-code), using
- OOA 3-folding and stacking with additional row [i] based on linear OOA(5122, 8388601, F5, 2, 13) (dual of [(8388601, 2), 16777080, 14]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(5122, 8388602, F5, 2, 13) (dual of [(8388602, 2), 16777082, 14]-NRT-code), using
- trace code [i] based on linear OOA(2561, 4194301, F25, 2, 13) (dual of [(4194301, 2), 8388541, 14]-NRT-code), using
- OOA 2-folding [i] based on linear OA(2561, 8388602, F25, 13) (dual of [8388602, 8388541, 14]-code), using
- discarding factors / shortening the dual code based on linear OA(2561, large, F25, 13) (dual of [large, large−61, 14]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 9765626 | 2510−1, defining interval I = [0,6], and minimum distance d ≥ |{−6,−5,…,6}|+1 = 14 (BCH-bound) [i]
- discarding factors / shortening the dual code based on linear OA(2561, large, F25, 13) (dual of [large, large−61, 14]-code), using
- OOA 2-folding [i] based on linear OA(2561, 8388602, F25, 13) (dual of [8388602, 8388541, 14]-code), using
- trace code [i] based on linear OOA(2561, 4194301, F25, 2, 13) (dual of [(4194301, 2), 8388541, 14]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(5122, 8388602, F5, 2, 13) (dual of [(8388602, 2), 16777082, 14]-NRT-code), using
- OOA 3-folding and stacking with additional row [i] based on linear OOA(5122, 8388601, F5, 2, 13) (dual of [(8388601, 2), 16777080, 14]-NRT-code), using
- net defined by OOA [i] based on linear OOA(5122, 2796200, F5, 14, 13) (dual of [(2796200, 14), 39146678, 14]-NRT-code), using
- digital (19, 25, 5208)-net over F5, using
(134, 134+13, large)-Net over F5 — Digital
Digital (134, 147, large)-net over F5, using
- 51 times duplication [i] based on digital (133, 146, large)-net over F5, using
- t-expansion [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
- t-expansion [i] based on digital (129, 146, large)-net over F5, using
(134, 134+13, large)-Net in Base 5 — Upper bound on s
There is no (134, 147, large)-net in base 5, because
- 11 times m-reduction [i] would yield (134, 136, large)-net in base 5, but