Best Known (116, 116+10, s)-Nets in Base 5
(116, 116+10, 3746068)-Net over F5 — Constructive and digital
Digital (116, 126, 3746068)-net over F5, using
- (u, u+v)-construction [i] based on
- digital (29, 34, 390628)-net over F5, using
- net defined by OOA [i] based on linear OOA(534, 390628, F5, 5, 5) (dual of [(390628, 5), 1953106, 6]-NRT-code), using
- OOA 2-folding and stacking with additional row [i] based on linear OA(534, 781257, F5, 5) (dual of [781257, 781223, 6]-code), using
- discarding factors / shortening the dual code based on linear OA(534, 781258, F5, 5) (dual of [781258, 781224, 6]-code), using
- trace code [i] based on linear OA(2517, 390629, F25, 5) (dual of [390629, 390612, 6]-code), using
- construction X applied to Ce(4) ⊂ Ce(3) [i] based on
- linear OA(2517, 390625, F25, 5) (dual of [390625, 390608, 6]-code), using an extension Ce(4) of the primitive narrow-sense BCH-code C(I) with length 390624 = 254−1, defining interval I = [1,4], and designed minimum distance d ≥ |I|+1 = 5 [i]
- linear OA(2513, 390625, F25, 4) (dual of [390625, 390612, 5]-code), using an extension Ce(3) of the primitive narrow-sense BCH-code C(I) with length 390624 = 254−1, defining interval I = [1,3], and designed minimum distance d ≥ |I|+1 = 4 [i]
- linear OA(250, 4, F25, 0) (dual of [4, 4, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(250, s, F25, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(4) ⊂ Ce(3) [i] based on
- trace code [i] based on linear OA(2517, 390629, F25, 5) (dual of [390629, 390612, 6]-code), using
- discarding factors / shortening the dual code based on linear OA(534, 781258, F5, 5) (dual of [781258, 781224, 6]-code), using
- OOA 2-folding and stacking with additional row [i] based on linear OA(534, 781257, F5, 5) (dual of [781257, 781223, 6]-code), using
- net defined by OOA [i] based on linear OOA(534, 390628, F5, 5, 5) (dual of [(390628, 5), 1953106, 6]-NRT-code), 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 (29, 34, 390628)-net over F5, using
(116, 116+10, large)-Net over F5 — Digital
Digital (116, 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
(116, 116+10, large)-Net in Base 5 — Upper bound on s
There is no (116, 126, large)-net in base 5, because
- 8 times m-reduction [i] would yield (116, 118, large)-net in base 5, but