Best Known (49, 60, s)-Nets in Base 5
(49, 60, 15628)-Net over F5 — Constructive and digital
Digital (49, 60, 15628)-net over F5, using
- net defined by OOA [i] based on linear OOA(560, 15628, F5, 11, 11) (dual of [(15628, 11), 171848, 12]-NRT-code), using
- OOA 5-folding and stacking with additional row [i] based on linear OA(560, 78141, F5, 11) (dual of [78141, 78081, 12]-code), using
- discarding factors / shortening the dual code based on linear OA(560, 78142, F5, 11) (dual of [78142, 78082, 12]-code), using
- construction X applied to Ce(10) ⊂ Ce(7) [i] based on
- linear OA(557, 78125, F5, 11) (dual of [78125, 78068, 12]-code), using an extension Ce(10) of the primitive narrow-sense BCH-code C(I) with length 78124 = 57−1, defining interval I = [1,10], and designed minimum distance d ≥ |I|+1 = 11 [i]
- linear OA(543, 78125, F5, 8) (dual of [78125, 78082, 9]-code), using an extension Ce(7) of the primitive narrow-sense BCH-code C(I) with length 78124 = 57−1, defining interval I = [1,7], and designed minimum distance d ≥ |I|+1 = 8 [i]
- linear OA(53, 17, F5, 2) (dual of [17, 14, 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(560, 78142, F5, 11) (dual of [78142, 78082, 12]-code), using
- OOA 5-folding and stacking with additional row [i] based on linear OA(560, 78141, F5, 11) (dual of [78141, 78081, 12]-code), using
(49, 60, 39606)-Net over F5 — Digital
Digital (49, 60, 39606)-net over F5, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(560, 39606, F5, 11) (dual of [39606, 39546, 12]-code), using
- discarding factors / shortening the dual code based on linear OA(560, 78132, F5, 11) (dual of [78132, 78072, 12]-code), using
- construction X applied to C([0,5]) ⊂ C([0,3]) [i] based on
- linear OA(557, 78126, F5, 11) (dual of [78126, 78069, 12]-code), using the expurgated narrow-sense BCH-code C(I) with length 78126 | 514−1, defining interval I = [0,5], and minimum distance d ≥ |{−5,−4,…,5}|+1 = 12 (BCH-bound) [i]
- linear OA(543, 78126, F5, 7) (dual of [78126, 78083, 8]-code), using the expurgated narrow-sense BCH-code C(I) with length 78126 | 514−1, 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(560, 78132, F5, 11) (dual of [78132, 78072, 12]-code), using
(49, 60, large)-Net in Base 5 — Upper bound on s
There is no (49, 60, large)-net in base 5, because
- 9 times m-reduction [i] would yield (49, 51, large)-net in base 5, but