Best Known (89, 89+10, s)-Nets in Base 7
(89, 89+10, 3355461)-Net over F7 — Constructive and digital
Digital (89, 99, 3355461)-net over F7, using
- (u, u+v)-construction [i] based on
- digital (2, 7, 21)-net over F7, using
- net defined by OOA [i] based on linear OOA(77, 21, F7, 5, 5) (dual of [(21, 5), 98, 6]-NRT-code), using
- OOA 2-folding and stacking with additional row [i] based on linear OA(77, 43, F7, 5) (dual of [43, 36, 6]-code), using
- net defined by OOA [i] based on linear OOA(77, 21, F7, 5, 5) (dual of [(21, 5), 98, 6]-NRT-code), using
- digital (82, 92, 3355440)-net over F7, using
- trace code for nets [i] based on digital (36, 46, 1677720)-net over F49, using
- net defined by OOA [i] based on linear OOA(4946, 1677720, F49, 10, 10) (dual of [(1677720, 10), 16777154, 11]-NRT-code), using
- OA 5-folding and stacking [i] based on linear OA(4946, 8388600, F49, 10) (dual of [8388600, 8388554, 11]-code), using
- discarding factors / shortening the dual code based on linear OA(4946, large, F49, 10) (dual of [large, large−46, 11]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 25679568 | 495−1, defining interval I = [0,9], and designed minimum distance d ≥ |I|+1 = 11 [i]
- discarding factors / shortening the dual code based on linear OA(4946, large, F49, 10) (dual of [large, large−46, 11]-code), using
- OA 5-folding and stacking [i] based on linear OA(4946, 8388600, F49, 10) (dual of [8388600, 8388554, 11]-code), using
- net defined by OOA [i] based on linear OOA(4946, 1677720, F49, 10, 10) (dual of [(1677720, 10), 16777154, 11]-NRT-code), using
- trace code for nets [i] based on digital (36, 46, 1677720)-net over F49, using
- digital (2, 7, 21)-net over F7, using
(89, 89+10, large)-Net over F7 — Digital
Digital (89, 99, large)-net over F7, using
- 71 times duplication [i] based on digital (88, 98, large)-net over F7, using
- t-expansion [i] based on digital (85, 98, large)-net over F7, using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(798, large, F7, 13) (dual of [large, large−98, 14]-code), using
- trace code [i] based on linear OA(4949, 5764802, F49, 13) (dual of [5764802, 5764753, 14]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 5764802 | 498−1, defining interval I = [0,6], and minimum distance d ≥ |{−6,−5,…,6}|+1 = 14 (BCH-bound) [i]
- trace code [i] based on linear OA(4949, 5764802, F49, 13) (dual of [5764802, 5764753, 14]-code), using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(798, large, F7, 13) (dual of [large, large−98, 14]-code), using
- t-expansion [i] based on digital (85, 98, large)-net over F7, using
(89, 89+10, large)-Net in Base 7 — Upper bound on s
There is no (89, 99, large)-net in base 7, because
- 8 times m-reduction [i] would yield (89, 91, large)-net in base 7, but