Best Known (115, 126, s)-Nets in Base 5
(115, 126, 3363251)-Net over F5 — Constructive and digital
Digital (115, 126, 3363251)-net over F5, using
- (u, u+v)-construction [i] based on
- digital (19, 24, 7811)-net over F5, using
- net defined by OOA [i] based on linear OOA(524, 7811, F5, 5, 5) (dual of [(7811, 5), 39031, 6]-NRT-code), using
- OOA 2-folding and stacking with additional row [i] based on linear OA(524, 15623, F5, 5) (dual of [15623, 15599, 6]-code), using
- discarding factors / shortening the dual code based on linear OA(524, 15624, F5, 5) (dual of [15624, 15600, 6]-code), using
- 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(524, 15624, F5, 5) (dual of [15624, 15600, 6]-code), using
- OOA 2-folding and stacking with additional row [i] based on linear OA(524, 15623, F5, 5) (dual of [15623, 15599, 6]-code), using
- net defined by OOA [i] based on linear OOA(524, 7811, F5, 5, 5) (dual of [(7811, 5), 39031, 6]-NRT-code), using
- digital (91, 102, 3355440)-net over F5, using
- trace code for nets [i] based on digital (40, 51, 1677720)-net over F25, using
- net defined by OOA [i] based on linear OOA(2551, 1677720, F25, 11, 11) (dual of [(1677720, 11), 18454869, 12]-NRT-code), using
- OOA 5-folding and stacking with additional row [i] based on linear OA(2551, 8388601, F25, 11) (dual of [8388601, 8388550, 12]-code), using
- discarding factors / shortening the dual code based on linear OA(2551, large, F25, 11) (dual of [large, large−51, 12]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 9765626 | 2510−1, defining interval I = [0,5], and minimum distance d ≥ |{−5,−4,…,5}|+1 = 12 (BCH-bound) [i]
- discarding factors / shortening the dual code based on linear OA(2551, large, F25, 11) (dual of [large, large−51, 12]-code), using
- OOA 5-folding and stacking with additional row [i] based on linear OA(2551, 8388601, F25, 11) (dual of [8388601, 8388550, 12]-code), using
- net defined by OOA [i] based on linear OOA(2551, 1677720, F25, 11, 11) (dual of [(1677720, 11), 18454869, 12]-NRT-code), using
- trace code for nets [i] based on digital (40, 51, 1677720)-net over F25, using
- digital (19, 24, 7811)-net over F5, using
(115, 126, large)-Net over F5 — Digital
Digital (115, 126, large)-net over F5, using
- t-expansion [i] based on digital (112, 126, large)-net over F5, using
- 1 times m-reduction [i] based on digital (112, 127, large)-net over F5, using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(5127, large, F5, 15) (dual of [large, large−127, 16]-code), using
- 7 times code embedding in larger space [i] based on linear OA(5120, large, F5, 15) (dual of [large, large−120, 16]-code), using
- the primitive narrow-sense BCH-code C(I) with length 9765624 = 510−1, defining interval I = [1,15], and designed minimum distance d ≥ |I|+1 = 16 [i]
- 7 times code embedding in larger space [i] based on linear OA(5120, large, F5, 15) (dual of [large, large−120, 16]-code), using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(5127, large, F5, 15) (dual of [large, large−127, 16]-code), using
- 1 times m-reduction [i] based on digital (112, 127, large)-net over F5, using
(115, 126, large)-Net in Base 5 — Upper bound on s
There is no (115, 126, large)-net in base 5, because
- 9 times m-reduction [i] would yield (115, 117, large)-net in base 5, but