Best Known (136, 136+12, s)-Nets in Base 5
(136, 136+12, 2926414)-Net over F5 — Constructive and digital
Digital (136, 148, 2926414)-net over F5, using
- (u, u+v)-construction [i] based on
- digital (30, 36, 130214)-net over F5, using
- net defined by OOA [i] based on linear OOA(536, 130214, F5, 6, 6) (dual of [(130214, 6), 781248, 7]-NRT-code), using
- OA 3-folding and stacking [i] based on linear OA(536, 390642, F5, 6) (dual of [390642, 390606, 7]-code), using
- discarding factors / shortening the dual code based on linear OA(536, 390644, F5, 6) (dual of [390644, 390608, 7]-code), using
- construction X applied to Ce(5) ⊂ Ce(2) [i] based on
- linear OA(533, 390625, F5, 6) (dual of [390625, 390592, 7]-code), using an extension Ce(5) of the primitive narrow-sense BCH-code C(I) with length 390624 = 58−1, defining interval I = [1,5], and designed minimum distance d ≥ |I|+1 = 6 [i]
- linear OA(517, 390625, F5, 3) (dual of [390625, 390608, 4]-code or 390625-cap in PG(16,5)), using an extension Ce(2) of the primitive narrow-sense BCH-code C(I) with length 390624 = 58−1, defining interval I = [1,2], and designed minimum distance d ≥ |I|+1 = 3 [i]
- linear OA(53, 19, F5, 2) (dual of [19, 16, 3]-code), using
- discarding factors / shortening the dual code based on linear OA(53, 24, F5, 2) (dual of [24, 21, 3]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 24 = 52−1, defining interval I = [0,1], and designed minimum distance d ≥ |I|+1 = 3 [i]
- discarding factors / shortening the dual code based on linear OA(53, 24, F5, 2) (dual of [24, 21, 3]-code), using
- construction X applied to Ce(5) ⊂ Ce(2) [i] based on
- discarding factors / shortening the dual code based on linear OA(536, 390644, F5, 6) (dual of [390644, 390608, 7]-code), using
- OA 3-folding and stacking [i] based on linear OA(536, 390642, F5, 6) (dual of [390642, 390606, 7]-code), using
- net defined by OOA [i] based on linear OOA(536, 130214, F5, 6, 6) (dual of [(130214, 6), 781248, 7]-NRT-code), using
- digital (100, 112, 2796200)-net over F5, using
- net defined by OOA [i] based on linear OOA(5112, 2796200, F5, 14, 12) (dual of [(2796200, 14), 39146688, 13]-NRT-code), using
- OOA 3-folding and stacking with additional row [i] based on linear OOA(5112, 8388601, F5, 2, 12) (dual of [(8388601, 2), 16777090, 13]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(5112, 8388602, F5, 2, 12) (dual of [(8388602, 2), 16777092, 13]-NRT-code), using
- trace code [i] based on linear OOA(2556, 4194301, F25, 2, 12) (dual of [(4194301, 2), 8388546, 13]-NRT-code), using
- OOA 2-folding [i] based on linear OA(2556, 8388602, F25, 12) (dual of [8388602, 8388546, 13]-code), using
- discarding factors / shortening the dual code based on linear OA(2556, large, F25, 12) (dual of [large, large−56, 13]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 9765624 = 255−1, defining interval I = [0,11], and designed minimum distance d ≥ |I|+1 = 13 [i]
- discarding factors / shortening the dual code based on linear OA(2556, large, F25, 12) (dual of [large, large−56, 13]-code), using
- OOA 2-folding [i] based on linear OA(2556, 8388602, F25, 12) (dual of [8388602, 8388546, 13]-code), using
- trace code [i] based on linear OOA(2556, 4194301, F25, 2, 12) (dual of [(4194301, 2), 8388546, 13]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(5112, 8388602, F5, 2, 12) (dual of [(8388602, 2), 16777092, 13]-NRT-code), using
- OOA 3-folding and stacking with additional row [i] based on linear OOA(5112, 8388601, F5, 2, 12) (dual of [(8388601, 2), 16777090, 13]-NRT-code), using
- net defined by OOA [i] based on linear OOA(5112, 2796200, F5, 14, 12) (dual of [(2796200, 14), 39146688, 13]-NRT-code), using
- digital (30, 36, 130214)-net over F5, using
(136, 136+12, large)-Net over F5 — Digital
Digital (136, 148, large)-net over F5, using
- 52 times duplication [i] based on digital (134, 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
(136, 136+12, large)-Net in Base 5 — Upper bound on s
There is no (136, 148, large)-net in base 5, because
- 10 times m-reduction [i] would yield (136, 138, large)-net in base 5, but