Best Known (6, 6+7, s)-Nets in Base 49
(6, 6+7, 800)-Net over F49 — Constructive and digital
Digital (6, 13, 800)-net over F49, using
- net defined by OOA [i] based on linear OOA(4913, 800, F49, 7, 7) (dual of [(800, 7), 5587, 8]-NRT-code), using
- OOA 3-folding and stacking with additional row [i] based on linear OA(4913, 2401, F49, 7) (dual of [2401, 2388, 8]-code), using
- an extension Ce(6) of the primitive narrow-sense BCH-code C(I) with length 2400 = 492−1, defining interval I = [1,6], and designed minimum distance d ≥ |I|+1 = 7 [i]
- OOA 3-folding and stacking with additional row [i] based on linear OA(4913, 2401, F49, 7) (dual of [2401, 2388, 8]-code), using
(6, 6+7, 1201)-Net over F49 — Digital
Digital (6, 13, 1201)-net over F49, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(4913, 1201, F49, 2, 7) (dual of [(1201, 2), 2389, 8]-NRT-code), using
- OOA 2-folding [i] based on linear OA(4913, 2402, F49, 7) (dual of [2402, 2389, 8]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 2402 | 494−1, defining interval I = [0,3], and minimum distance d ≥ |{−3,−2,…,3}|+1 = 8 (BCH-bound) [i]
- OOA 2-folding [i] based on linear OA(4913, 2402, F49, 7) (dual of [2402, 2389, 8]-code), using
(6, 6+7, 218235)-Net in Base 49 — Upper bound on s
There is no (6, 13, 218236)-net in base 49, because
- 1 times m-reduction [i] would yield (6, 12, 218236)-net in base 49, but
- the generalized Rao bound for nets shows that 49m ≥ 191 583447 399307 159489 > 4912 [i]