Best Known (19, 31, s)-Nets in Base 49
(19, 31, 452)-Net over F49 — Constructive and digital
Digital (19, 31, 452)-net over F49, using
- (u, u+v)-construction [i] based on
- digital (2, 8, 52)-net over F49, using
- net from sequence [i] based on digital (2, 51)-sequence over F49, using
- digital (11, 23, 400)-net over F49, using
- net defined by OOA [i] based on linear OOA(4923, 400, F49, 12, 12) (dual of [(400, 12), 4777, 13]-NRT-code), using
- OA 6-folding and stacking [i] based on linear OA(4923, 2400, F49, 12) (dual of [2400, 2377, 13]-code), using
- discarding factors / shortening the dual code based on linear OA(4923, 2401, F49, 12) (dual of [2401, 2378, 13]-code), using
- an extension Ce(11) of the primitive narrow-sense BCH-code C(I) with length 2400 = 492−1, defining interval I = [1,11], and designed minimum distance d ≥ |I|+1 = 12 [i]
- discarding factors / shortening the dual code based on linear OA(4923, 2401, F49, 12) (dual of [2401, 2378, 13]-code), using
- OA 6-folding and stacking [i] based on linear OA(4923, 2400, F49, 12) (dual of [2400, 2377, 13]-code), using
- net defined by OOA [i] based on linear OOA(4923, 400, F49, 12, 12) (dual of [(400, 12), 4777, 13]-NRT-code), using
- digital (2, 8, 52)-net over F49, using
(19, 31, 5939)-Net over F49 — Digital
Digital (19, 31, 5939)-net over F49, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4931, 5939, F49, 12) (dual of [5939, 5908, 13]-code), using
- 3528 step Varšamov–Edel lengthening with (ri) = (3, 6 times 0, 1, 41 times 0, 1, 162 times 0, 1, 496 times 0, 1, 1080 times 0, 1, 1737 times 0) [i] based on linear OA(4923, 2403, F49, 12) (dual of [2403, 2380, 13]-code), using
- construction X applied to Ce(11) ⊂ Ce(10) [i] based on
- linear OA(4923, 2401, F49, 12) (dual of [2401, 2378, 13]-code), using an extension Ce(11) of the primitive narrow-sense BCH-code C(I) with length 2400 = 492−1, defining interval I = [1,11], and designed minimum distance d ≥ |I|+1 = 12 [i]
- linear OA(4921, 2401, F49, 11) (dual of [2401, 2380, 12]-code), using an extension Ce(10) of the primitive narrow-sense BCH-code C(I) with length 2400 = 492−1, defining interval I = [1,10], and designed minimum distance d ≥ |I|+1 = 11 [i]
- linear OA(490, 2, F49, 0) (dual of [2, 2, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(490, s, F49, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(11) ⊂ Ce(10) [i] based on
- 3528 step Varšamov–Edel lengthening with (ri) = (3, 6 times 0, 1, 41 times 0, 1, 162 times 0, 1, 496 times 0, 1, 1080 times 0, 1, 1737 times 0) [i] based on linear OA(4923, 2403, F49, 12) (dual of [2403, 2380, 13]-code), using
(19, 31, large)-Net in Base 49 — Upper bound on s
There is no (19, 31, large)-net in base 49, because
- 10 times m-reduction [i] would yield (19, 21, large)-net in base 49, but