Best Known (45, 56, s)-Nets in Base 5
(45, 56, 3146)-Net over F5 — Constructive and digital
Digital (45, 56, 3146)-net over F5, using
- (u, u+v)-construction [i] based on
- digital (2, 7, 21)-net over F5, using
- 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
- net defined by OOA [i] based on linear OOA(549, 3125, F5, 11, 11) (dual of [(3125, 11), 34326, 12]-NRT-code), using
(45, 56, 15657)-Net over F5 — Digital
Digital (45, 56, 15657)-net over F5, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(556, 15657, F5, 11) (dual of [15657, 15601, 12]-code), using
- construction X with Varšamov bound [i] based on
- linear OA(555, 15655, F5, 11) (dual of [15655, 15600, 12]-code), using
- construction X applied to Ce(10) ⊂ Ce(5) [i] based on
- linear OA(549, 15625, F5, 11) (dual of [15625, 15576, 12]-code), using an extension Ce(10) of the primitive narrow-sense BCH-code C(I) with length 15624 = 56−1, defining interval I = [1,10], and designed minimum distance d ≥ |I|+1 = 11 [i]
- linear OA(525, 15625, F5, 6) (dual of [15625, 15600, 7]-code), using an extension Ce(5) of the primitive narrow-sense BCH-code C(I) with length 15624 = 56−1, defining interval I = [1,5], and designed minimum distance d ≥ |I|+1 = 6 [i]
- linear OA(56, 30, F5, 4) (dual of [30, 24, 5]-code), using
- construction X applied to Ce(10) ⊂ Ce(5) [i] based on
- linear OA(555, 15656, F5, 10) (dual of [15656, 15601, 11]-code), using Gilbert–Varšamov bound and bm = 555 > Vbs−1(k−1) = 40 711594 561041 026427 089149 653605 443245 [i]
- linear OA(50, 1, F5, 0) (dual of [1, 1, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(50, s, F5, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- linear OA(555, 15655, F5, 11) (dual of [15655, 15600, 12]-code), using
- construction X with Varšamov bound [i] based on
(45, 56, large)-Net in Base 5 — Upper bound on s
There is no (45, 56, large)-net in base 5, because
- 9 times m-reduction [i] would yield (45, 47, large)-net in base 5, but