Best Known (50−10, 50, s)-Nets in Base 5
(50−10, 50, 3127)-Net over F5 — Constructive and digital
Digital (40, 50, 3127)-net over F5, using
- net defined by OOA [i] based on linear OOA(550, 3127, F5, 10, 10) (dual of [(3127, 10), 31220, 11]-NRT-code), using
- OA 5-folding and stacking [i] based on linear OA(550, 15635, F5, 10) (dual of [15635, 15585, 11]-code), using
- discarding factors / shortening the dual code based on linear OA(550, 15638, F5, 10) (dual of [15638, 15588, 11]-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(51, 13, F5, 1) (dual of [13, 12, 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 X applied to Ce(10) ⊂ Ce(7) [i] based on
- discarding factors / shortening the dual code based on linear OA(550, 15638, F5, 10) (dual of [15638, 15588, 11]-code), using
- OA 5-folding and stacking [i] based on linear OA(550, 15635, F5, 10) (dual of [15635, 15585, 11]-code), using
(50−10, 50, 15639)-Net over F5 — Digital
Digital (40, 50, 15639)-net over F5, using
- embedding of OOA with Gilbert–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
(50−10, 50, 6360277)-Net in Base 5 — Upper bound on s
There is no (40, 50, 6360278)-net in base 5, because
- the generalized Rao bound for nets shows that 5m ≥ 88817 861624 978277 072233 460482 736345 > 550 [i]