Best Known (109−15, 109, s)-Nets in Base 7
(109−15, 109, 1198371)-Net over F7 — Constructive and digital
Digital (94, 109, 1198371)-net over F7, using
- net defined by OOA [i] based on linear OOA(7109, 1198371, F7, 15, 15) (dual of [(1198371, 15), 17975456, 16]-NRT-code), using
- OOA 7-folding and stacking with additional row [i] based on linear OA(7109, 8388598, F7, 15) (dual of [8388598, 8388489, 16]-code), using
- discarding factors / shortening the dual code 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]
- discarding factors / shortening the dual code based on linear OA(7109, large, F7, 15) (dual of [large, large−109, 16]-code), using
- OOA 7-folding and stacking with additional row [i] based on linear OA(7109, 8388598, F7, 15) (dual of [8388598, 8388489, 16]-code), using
(109−15, 109, large)-Net over F7 — Digital
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]
(109−15, 109, large)-Net in Base 7 — Upper bound on s
There is no (94, 109, large)-net in base 7, because
- 13 times m-reduction [i] would yield (94, 96, large)-net in base 7, but