Best Known (48−10, 48, s)-Nets in Base 5
(48−10, 48, 3125)-Net over F5 — Constructive and digital
Digital (38, 48, 3125)-net over F5, using
- net defined by OOA [i] based on linear OOA(548, 3125, F5, 10, 10) (dual of [(3125, 10), 31202, 11]-NRT-code), using
- OA 5-folding and stacking [i] based on linear OA(548, 15625, F5, 10) (dual of [15625, 15577, 11]-code), using
- 1 times truncation [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]
- 1 times truncation [i] based on linear OA(549, 15626, F5, 11) (dual of [15626, 15577, 12]-code), using
- OA 5-folding and stacking [i] based on linear OA(548, 15625, F5, 10) (dual of [15625, 15577, 11]-code), using
(48−10, 48, 12020)-Net over F5 — Digital
Digital (38, 48, 12020)-net over F5, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(548, 12020, F5, 10) (dual of [12020, 11972, 11]-code), using
- discarding factors / shortening the dual code based on linear OA(548, 15625, F5, 10) (dual of [15625, 15577, 11]-code), using
- 1 times truncation [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]
- 1 times truncation [i] based on linear OA(549, 15626, F5, 11) (dual of [15626, 15577, 12]-code), using
- discarding factors / shortening the dual code based on linear OA(548, 15625, F5, 10) (dual of [15625, 15577, 11]-code), using
(48−10, 48, 3341087)-Net in Base 5 — Upper bound on s
There is no (38, 48, 3341088)-net in base 5, because
- the generalized Rao bound for nets shows that 5m ≥ 3552 715212 572208 116697 092482 084993 > 548 [i]