Best Known (102, 102+10, s)-Nets in Base 5
(102, 102+10, 3357242)-Net over F5 — Constructive and digital
Digital (102, 112, 3357242)-net over F5, using
- (u, u+v)-construction [i] based on
- digital (15, 20, 1802)-net over F5, using
- net defined by OOA [i] based on linear OOA(520, 1802, F5, 6, 5) (dual of [(1802, 6), 10792, 6]-NRT-code), using
- OOA stacking with additional row [i] based on linear OOA(520, 1803, F5, 2, 5) (dual of [(1803, 2), 3586, 6]-NRT-code), using
- generalized (u, u+v)-construction [i] based on
- linear OOA(51, 601, F5, 2, 1) (dual of [(601, 2), 1201, 2]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(51, s, F5, 2, 1) with arbitrarily large s, using
- appending 1 arbitrary column [i] based on linear OA(51, s, F5, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- discarding factors / shortening the dual code based on linear OOA(51, s, F5, 2, 1) with arbitrarily large s, using
- linear OOA(55, 601, F5, 2, 2) (dual of [(601, 2), 1197, 3]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(55, 781, F5, 2, 2) (dual of [(781, 2), 1557, 3]-NRT-code), using
- appending kth column [i] based on linear OA(55, 781, F5, 2) (dual of [781, 776, 3]-code), using
- Hamming code H(5,5) [i]
- appending kth column [i] based on linear OA(55, 781, F5, 2) (dual of [781, 776, 3]-code), using
- discarding factors / shortening the dual code based on linear OOA(55, 781, F5, 2, 2) (dual of [(781, 2), 1557, 3]-NRT-code), using
- linear OOA(514, 601, F5, 2, 5) (dual of [(601, 2), 1188, 6]-NRT-code), using
- OOA 2-folding [i] based on linear OA(514, 1202, F5, 5) (dual of [1202, 1188, 6]-code), using
- trace code [i] based on linear OA(257, 601, F25, 5) (dual of [601, 594, 6]-code), using
- OOA 2-folding [i] based on linear OA(514, 1202, F5, 5) (dual of [1202, 1188, 6]-code), using
- linear OOA(51, 601, F5, 2, 1) (dual of [(601, 2), 1201, 2]-NRT-code), using
- generalized (u, u+v)-construction [i] based on
- OOA stacking with additional row [i] based on linear OOA(520, 1803, F5, 2, 5) (dual of [(1803, 2), 3586, 6]-NRT-code), using
- net defined by OOA [i] based on linear OOA(520, 1802, F5, 6, 5) (dual of [(1802, 6), 10792, 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 (15, 20, 1802)-net over F5, using
(102, 102+10, large)-Net over F5 — Digital
Digital (102, 112, large)-net over F5, using
- 53 times duplication [i] based on digital (99, 109, large)-net over F5, using
- t-expansion [i] based on digital (96, 109, large)-net over F5, using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(5109, large, F5, 13) (dual of [large, large−109, 14]-code), using
- 8 times code embedding in larger space [i] based on linear OA(5101, large, F5, 13) (dual of [large, large−101, 14]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 9765626 | 520−1, defining interval I = [0,6], and minimum distance d ≥ |{−6,−5,…,6}|+1 = 14 (BCH-bound) [i]
- 8 times code embedding in larger space [i] based on linear OA(5101, large, F5, 13) (dual of [large, large−101, 14]-code), using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(5109, large, F5, 13) (dual of [large, large−109, 14]-code), using
- t-expansion [i] based on digital (96, 109, large)-net over F5, using
(102, 102+10, large)-Net in Base 5 — Upper bound on s
There is no (102, 112, large)-net in base 5, because
- 8 times m-reduction [i] would yield (102, 104, large)-net in base 5, but