Best Known (49−15, 49, s)-Nets in Base 7
(49−15, 49, 343)-Net over F7 — Constructive and digital
Digital (34, 49, 343)-net over F7, using
- net defined by OOA [i] based on linear OOA(749, 343, F7, 15, 15) (dual of [(343, 15), 5096, 16]-NRT-code), using
- OOA 7-folding and stacking with additional row [i] based on linear OA(749, 2402, F7, 15) (dual of [2402, 2353, 16]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 2402 | 78−1, defining interval I = [0,7], and minimum distance d ≥ |{−7,−6,…,7}|+1 = 16 (BCH-bound) [i]
- OOA 7-folding and stacking with additional row [i] based on linear OA(749, 2402, F7, 15) (dual of [2402, 2353, 16]-code), using
(49−15, 49, 1239)-Net over F7 — Digital
Digital (34, 49, 1239)-net over F7, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(749, 1239, F7, 15) (dual of [1239, 1190, 16]-code), using
- discarding factors / shortening the dual code based on linear OA(749, 2401, F7, 15) (dual of [2401, 2352, 16]-code), using
- an extension Ce(14) of the primitive narrow-sense BCH-code C(I) with length 2400 = 74−1, defining interval I = [1,14], and designed minimum distance d ≥ |I|+1 = 15 [i]
- discarding factors / shortening the dual code based on linear OA(749, 2401, F7, 15) (dual of [2401, 2352, 16]-code), using
(49−15, 49, 351334)-Net in Base 7 — Upper bound on s
There is no (34, 49, 351335)-net in base 7, because
- 1 times m-reduction [i] would yield (34, 48, 351335)-net in base 7, but
- the generalized Rao bound for nets shows that 7m ≥ 36704 066592 959811 893020 203707 037155 740183 > 748 [i]