Best Known (39−8, 39, s)-Nets in Base 5
(39−8, 39, 3910)-Net over F5 — Constructive and digital
Digital (31, 39, 3910)-net over F5, using
- net defined by OOA [i] based on linear OOA(539, 3910, F5, 8, 8) (dual of [(3910, 8), 31241, 9]-NRT-code), using
- OA 4-folding and stacking [i] based on linear OA(539, 15640, F5, 8) (dual of [15640, 15601, 9]-code), using
- 1 times code embedding in larger space [i] based on linear OA(538, 15639, F5, 8) (dual of [15639, 15601, 9]-code), using
- construction X4 applied to Ce(7) ⊂ Ce(5) [i] based on
- 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(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(513, 14, F5, 13) (dual of [14, 1, 14]-code or 14-arc in PG(12,5)), using
- dual of repetition code with length 14 [i]
- linear OA(51, 14, F5, 1) (dual of [14, 13, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(51, s, F5, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X4 applied to Ce(7) ⊂ Ce(5) [i] based on
- 1 times code embedding in larger space [i] based on linear OA(538, 15639, F5, 8) (dual of [15639, 15601, 9]-code), using
- OA 4-folding and stacking [i] based on linear OA(539, 15640, F5, 8) (dual of [15640, 15601, 9]-code), using
(39−8, 39, 15641)-Net over F5 — Digital
Digital (31, 39, 15641)-net over F5, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(539, 15641, F5, 8) (dual of [15641, 15602, 9]-code), using
- construction X with Varšamov bound [i] based on
- linear OA(538, 15639, F5, 8) (dual of [15639, 15601, 9]-code), using
- construction X4 applied to Ce(7) ⊂ Ce(5) [i] based on
- 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(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(513, 14, F5, 13) (dual of [14, 1, 14]-code or 14-arc in PG(12,5)), using
- dual of repetition code with length 14 [i]
- linear OA(51, 14, F5, 1) (dual of [14, 13, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(51, s, F5, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X4 applied to Ce(7) ⊂ Ce(5) [i] based on
- linear OA(538, 15640, F5, 7) (dual of [15640, 15602, 8]-code), using Gilbert–Varšamov bound and bm = 538 > Vbs−1(k−1) = 83 158452 840508 053832 593901 [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(538, 15639, F5, 8) (dual of [15639, 15601, 9]-code), using
- construction X with Varšamov bound [i] based on
(39−8, 39, 3613683)-Net in Base 5 — Upper bound on s
There is no (31, 39, 3613684)-net in base 5, because
- the generalized Rao bound for nets shows that 5m ≥ 1818 991278 445186 989639 674625 > 539 [i]