Best Known (122−10, 122, s)-Nets in Base 5
(122−10, 122, 3394510)-Net over F5 — Constructive and digital
Digital (112, 122, 3394510)-net over F5, using
- (u, u+v)-construction [i] based on
- digital (25, 30, 39070)-net over F5, using
- net defined by OOA [i] based on linear OOA(530, 39070, F5, 5, 5) (dual of [(39070, 5), 195320, 6]-NRT-code), using
- OOA 2-folding and stacking with additional row [i] based on linear OA(530, 78141, F5, 5) (dual of [78141, 78111, 6]-code), using
- construction X applied to C([0,2]) ⊂ C([0,1]) [i] based on
- linear OA(529, 78126, F5, 5) (dual of [78126, 78097, 6]-code), using the expurgated narrow-sense BCH-code C(I) with length 78126 | 514−1, defining interval I = [0,2], and minimum distance d ≥ |{−2,−1,0,1,2}|+1 = 6 (BCH-bound) [i]
- linear OA(515, 78126, F5, 3) (dual of [78126, 78111, 4]-code or 78126-cap in PG(14,5)), using the expurgated narrow-sense BCH-code C(I) with length 78126 | 514−1, defining interval I = [0,1], and minimum distance d ≥ |{−1,0,1}|+1 = 4 (BCH-bound) [i]
- linear OA(51, 15, F5, 1) (dual of [15, 14, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(51, s, F5, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to C([0,2]) ⊂ C([0,1]) [i] based on
- OOA 2-folding and stacking with additional row [i] based on linear OA(530, 78141, F5, 5) (dual of [78141, 78111, 6]-code), using
- net defined by OOA [i] based on linear OOA(530, 39070, F5, 5, 5) (dual of [(39070, 5), 195320, 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 (25, 30, 39070)-net over F5, using
(122−10, 122, large)-Net over F5 — Digital
Digital (112, 122, large)-net over F5, using
- 5 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
(122−10, 122, large)-Net in Base 5 — Upper bound on s
There is no (112, 122, large)-net in base 5, because
- 8 times m-reduction [i] would yield (112, 114, large)-net in base 5, but