Best Known (51−10, 51, s)-Nets in Base 5
(51−10, 51, 3128)-Net over F5 — Constructive and digital
Digital (41, 51, 3128)-net over F5, using
- net defined by OOA [i] based on linear OOA(551, 3128, F5, 10, 10) (dual of [(3128, 10), 31229, 11]-NRT-code), using
- OA 5-folding and stacking [i] based on linear OA(551, 15640, F5, 10) (dual of [15640, 15589, 11]-code), using
- 1 times code embedding in larger space [i] based on linear OA(550, 15639, F5, 10) (dual of [15639, 15589, 11]-code), using
- construction X4 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(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(10) ⊂ Ce(7) [i] based on
- 1 times code embedding in larger space [i] based on linear OA(550, 15639, F5, 10) (dual of [15639, 15589, 11]-code), using
- OA 5-folding and stacking [i] based on linear OA(551, 15640, F5, 10) (dual of [15640, 15589, 11]-code), using
(51−10, 51, 15641)-Net over F5 — Digital
Digital (41, 51, 15641)-net over F5, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(551, 15641, F5, 10) (dual of [15641, 15590, 11]-code), using
- construction X with Varšamov bound [i] based on
- linear OA(550, 15639, F5, 10) (dual of [15639, 15589, 11]-code), using
- construction X4 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(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(10) ⊂ Ce(7) [i] based on
- linear OA(550, 15640, F5, 9) (dual of [15640, 15590, 10]-code), using Gilbert–Varšamov bound and bm = 550 > Vbs−1(k−1) = 5806 429114 305176 921704 526252 028397 [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(550, 15639, F5, 10) (dual of [15639, 15589, 11]-code), using
- construction X with Varšamov bound [i] based on
(51−10, 51, large)-Net in Base 5 — Upper bound on s
There is no (41, 51, large)-net in base 5, because
- 8 times m-reduction [i] would yield (41, 43, large)-net in base 5, but