Best Known (41, 52, s)-Nets in Base 5
(41, 52, 3127)-Net over F5 — Constructive and digital
Digital (41, 52, 3127)-net over F5, using
- net defined by OOA [i] based on linear OOA(552, 3127, F5, 11, 11) (dual of [(3127, 11), 34345, 12]-NRT-code), using
- OOA 5-folding and stacking with additional row [i] based on linear OA(552, 15636, F5, 11) (dual of [15636, 15584, 12]-code), using
- discarding factors / shortening the dual code based on linear OA(552, 15640, F5, 11) (dual of [15640, 15588, 12]-code), using
- construction X applied to Ce(10) ⊂ Ce(7) [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(537, 15625, F5, 8) (dual of [15625, 15588, 9]-code), using an extension Ce(7) of the primitive narrow-sense BCH-code C(I) with length 15624 = 56−1, defining interval I = [1,7], and designed minimum distance d ≥ |I|+1 = 8 [i]
- linear OA(53, 15, F5, 2) (dual of [15, 12, 3]-code), using
- discarding factors / shortening the dual code based on linear OA(53, 24, F5, 2) (dual of [24, 21, 3]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 24 = 52−1, defining interval I = [0,1], and designed minimum distance d ≥ |I|+1 = 3 [i]
- discarding factors / shortening the dual code based on linear OA(53, 24, F5, 2) (dual of [24, 21, 3]-code), using
- construction X applied to Ce(10) ⊂ Ce(7) [i] based on
- discarding factors / shortening the dual code based on linear OA(552, 15640, F5, 11) (dual of [15640, 15588, 12]-code), using
- OOA 5-folding and stacking with additional row [i] based on linear OA(552, 15636, F5, 11) (dual of [15636, 15584, 12]-code), using
(41, 52, 9468)-Net over F5 — Digital
Digital (41, 52, 9468)-net over F5, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(552, 9468, F5, 11) (dual of [9468, 9416, 12]-code), using
- discarding factors / shortening the dual code based on linear OA(552, 15632, F5, 11) (dual of [15632, 15580, 12]-code), using
- construction X applied to C([0,5]) ⊂ C([0,3]) [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]
- linear OA(537, 15626, F5, 7) (dual of [15626, 15589, 8]-code), using the expurgated narrow-sense BCH-code C(I) with length 15626 | 512−1, defining interval I = [0,3], and minimum distance d ≥ |{−3,−2,…,3}|+1 = 8 (BCH-bound) [i]
- linear OA(53, 6, F5, 3) (dual of [6, 3, 4]-code or 6-arc in PG(2,5) or 6-cap in PG(2,5)), using
- extended Reed–Solomon code RSe(3,5) [i]
- oval in PG(2, 5) [i]
- construction X applied to C([0,5]) ⊂ C([0,3]) [i] based on
- discarding factors / shortening the dual code based on linear OA(552, 15632, F5, 11) (dual of [15632, 15580, 12]-code), using
(41, 52, large)-Net in Base 5 — Upper bound on s
There is no (41, 52, large)-net in base 5, because
- 9 times m-reduction [i] would yield (41, 43, large)-net in base 5, but