Best Known (98, 98+11, s)-Nets in Base 7
(98, 98+11, 3355461)-Net over F7 — Constructive and digital
Digital (98, 109, 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 (91, 102, 3355440)-net over F7, using
- trace code for nets [i] based on digital (40, 51, 1677720)-net over F49, using
- net defined by OOA [i] based on linear OOA(4951, 1677720, F49, 11, 11) (dual of [(1677720, 11), 18454869, 12]-NRT-code), using
- OOA 5-folding and stacking with additional row [i] based on linear OA(4951, 8388601, F49, 11) (dual of [8388601, 8388550, 12]-code), using
- discarding factors / shortening the dual code based on linear OA(4951, large, F49, 11) (dual of [large, large−51, 12]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 28247525 | 4910−1, defining interval I = [0,5], and minimum distance d ≥ |{−5,−4,…,5}|+1 = 12 (BCH-bound) [i]
- discarding factors / shortening the dual code based on linear OA(4951, large, F49, 11) (dual of [large, large−51, 12]-code), using
- OOA 5-folding and stacking with additional row [i] based on linear OA(4951, 8388601, F49, 11) (dual of [8388601, 8388550, 12]-code), using
- net defined by OOA [i] based on linear OOA(4951, 1677720, F49, 11, 11) (dual of [(1677720, 11), 18454869, 12]-NRT-code), using
- trace code for nets [i] based on digital (40, 51, 1677720)-net over F49, using
- digital (2, 7, 21)-net over F7, using
(98, 98+11, large)-Net over F7 — Digital
Digital (98, 109, large)-net over F7, using
- t-expansion [i] based on digital (94, 109, large)-net over F7, using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(7109, large, F7, 15) (dual of [large, large−109, 16]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 20176804 | 718−1, defining interval I = [0,7], and minimum distance d ≥ |{−7,−6,…,7}|+1 = 16 (BCH-bound) [i]
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(7109, large, F7, 15) (dual of [large, large−109, 16]-code), using
(98, 98+11, large)-Net in Base 7 — Upper bound on s
There is no (98, 109, large)-net in base 7, because
- 9 times m-reduction [i] would yield (98, 100, large)-net in base 7, but