Best Known (108, 118, s)-Nets in Base 5
(108, 118, 3371067)-Net over F5 — Constructive and digital
Digital (108, 118, 3371067)-net over F5, using
- (u, u+v)-construction [i] based on
- digital (21, 26, 15627)-net over F5, using
- net defined by OOA [i] based on linear OOA(526, 15627, F5, 5, 5) (dual of [(15627, 5), 78109, 6]-NRT-code), using
- OOA 2-folding and stacking with additional row [i] based on linear OA(526, 31255, F5, 5) (dual of [31255, 31229, 6]-code), using
- discarding factors / shortening the dual code based on linear OA(526, 31256, F5, 5) (dual of [31256, 31230, 6]-code), using
- trace code [i] based on linear OA(2513, 15628, F25, 5) (dual of [15628, 15615, 6]-code), using
- construction X applied to Ce(4) ⊂ Ce(3) [i] based on
- linear OA(2513, 15625, F25, 5) (dual of [15625, 15612, 6]-code), using an extension Ce(4) of the primitive narrow-sense BCH-code C(I) with length 15624 = 253−1, defining interval I = [1,4], and designed minimum distance d ≥ |I|+1 = 5 [i]
- linear OA(2510, 15625, F25, 4) (dual of [15625, 15615, 5]-code), using an extension Ce(3) of the primitive narrow-sense BCH-code C(I) with length 15624 = 253−1, defining interval I = [1,3], and designed minimum distance d ≥ |I|+1 = 4 [i]
- linear OA(250, 3, F25, 0) (dual of [3, 3, 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(2513, 15628, F25, 5) (dual of [15628, 15615, 6]-code), using
- discarding factors / shortening the dual code based on linear OA(526, 31256, F5, 5) (dual of [31256, 31230, 6]-code), using
- OOA 2-folding and stacking with additional row [i] based on linear OA(526, 31255, F5, 5) (dual of [31255, 31229, 6]-code), using
- net defined by OOA [i] based on linear OOA(526, 15627, F5, 5, 5) (dual of [(15627, 5), 78109, 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 (21, 26, 15627)-net over F5, using
(108, 118, large)-Net over F5 — Digital
Digital (108, 118, large)-net over F5, using
- t-expansion [i] based on digital (104, 118, large)-net over F5, using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(5118, large, F5, 14) (dual of [large, large−118, 15]-code), using
- 7 times code embedding in larger space [i] based on linear OA(5111, large, F5, 14) (dual of [large, large−111, 15]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 9765624 = 510−1, defining interval I = [0,13], and designed minimum distance d ≥ |I|+1 = 15 [i]
- 7 times code embedding in larger space [i] based on linear OA(5111, large, F5, 14) (dual of [large, large−111, 15]-code), using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(5118, large, F5, 14) (dual of [large, large−118, 15]-code), using
(108, 118, large)-Net in Base 5 — Upper bound on s
There is no (108, 118, large)-net in base 5, because
- 8 times m-reduction [i] would yield (108, 110, large)-net in base 5, but