Best Known (47, 56, s)-Nets in Base 5
(47, 56, 19559)-Net over F5 — Constructive and digital
Digital (47, 56, 19559)-net over F5, using
- (u, u+v)-construction [i] based on
- digital (2, 6, 27)-net over F5, using
- digital (41, 50, 19532)-net over F5, using
- net defined by OOA [i] based on linear OOA(550, 19532, F5, 9, 9) (dual of [(19532, 9), 175738, 10]-NRT-code), using
- OOA 4-folding and stacking with additional row [i] based on linear OA(550, 78129, F5, 9) (dual of [78129, 78079, 10]-code), using
- discarding factors / shortening the dual code based on linear OA(550, 78132, F5, 9) (dual of [78132, 78082, 10]-code), using
- construction X applied to Ce(8) ⊂ Ce(7) [i] based on
- linear OA(550, 78125, F5, 9) (dual of [78125, 78075, 10]-code), using an extension Ce(8) of the primitive narrow-sense BCH-code C(I) with length 78124 = 57−1, defining interval I = [1,8], and designed minimum distance d ≥ |I|+1 = 9 [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(50, 7, F5, 0) (dual of [7, 7, 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
- construction X applied to Ce(8) ⊂ Ce(7) [i] based on
- discarding factors / shortening the dual code based on linear OA(550, 78132, F5, 9) (dual of [78132, 78082, 10]-code), using
- OOA 4-folding and stacking with additional row [i] based on linear OA(550, 78129, F5, 9) (dual of [78129, 78079, 10]-code), using
- net defined by OOA [i] based on linear OOA(550, 19532, F5, 9, 9) (dual of [(19532, 9), 175738, 10]-NRT-code), using
(47, 56, 78162)-Net over F5 — Digital
Digital (47, 56, 78162)-net over F5, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(556, 78162, F5, 9) (dual of [78162, 78106, 10]-code), using
- (u, u+v)-construction [i] based on
- linear OA(56, 30, F5, 4) (dual of [30, 24, 5]-code), using
- linear OA(550, 78132, F5, 9) (dual of [78132, 78082, 10]-code), using
- construction X applied to Ce(8) ⊂ Ce(7) [i] based on
- linear OA(550, 78125, F5, 9) (dual of [78125, 78075, 10]-code), using an extension Ce(8) of the primitive narrow-sense BCH-code C(I) with length 78124 = 57−1, defining interval I = [1,8], and designed minimum distance d ≥ |I|+1 = 9 [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(50, 7, F5, 0) (dual of [7, 7, 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
- construction X applied to Ce(8) ⊂ Ce(7) [i] based on
- (u, u+v)-construction [i] based on
(47, 56, large)-Net in Base 5 — Upper bound on s
There is no (47, 56, large)-net in base 5, because
- 7 times m-reduction [i] would yield (47, 49, large)-net in base 5, but