Best Known (49−11, 49, s)-Nets in Base 5
(49−11, 49, 3125)-Net over F5 — Constructive and digital
Digital (38, 49, 3125)-net over F5, using
- net defined by OOA [i] based on linear OOA(549, 3125, F5, 11, 11) (dual of [(3125, 11), 34326, 12]-NRT-code), using
- OOA 5-folding and stacking with additional row [i] based on linear OA(549, 15626, F5, 11) (dual of [15626, 15577, 12]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 15626 | 512−1, defining interval I = [0,5], and minimum distance d ≥ |{−5,−4,…,5}|+1 = 12 (BCH-bound) [i]
- OOA 5-folding and stacking with additional row [i] based on linear OA(549, 15626, F5, 11) (dual of [15626, 15577, 12]-code), using
(49−11, 49, 7813)-Net over F5 — Digital
Digital (38, 49, 7813)-net over F5, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(549, 7813, F5, 2, 11) (dual of [(7813, 2), 15577, 12]-NRT-code), using
- OOA 2-folding [i] based on linear OA(549, 15626, F5, 11) (dual of [15626, 15577, 12]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 15626 | 512−1, defining interval I = [0,5], and minimum distance d ≥ |{−5,−4,…,5}|+1 = 12 (BCH-bound) [i]
- OOA 2-folding [i] based on linear OA(549, 15626, F5, 11) (dual of [15626, 15577, 12]-code), using
(49−11, 49, 3341087)-Net in Base 5 — Upper bound on s
There is no (38, 49, 3341088)-net in base 5, because
- 1 times m-reduction [i] would yield (38, 48, 3341088)-net in base 5, but
- the generalized Rao bound for nets shows that 5m ≥ 3552 715212 572208 116697 092482 084993 > 548 [i]