Best Known (20, 20+13, s)-Nets in Base 49
(20, 20+13, 452)-Net over F49 — Constructive and digital
Digital (20, 33, 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 (12, 25, 400)-net over F49, using
- net defined by OOA [i] based on linear OOA(4925, 400, F49, 13, 13) (dual of [(400, 13), 5175, 14]-NRT-code), using
- OOA 6-folding and stacking with additional row [i] based on linear OA(4925, 2401, F49, 13) (dual of [2401, 2376, 14]-code), using
- an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 2400 = 492−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- OOA 6-folding and stacking with additional row [i] based on linear OA(4925, 2401, F49, 13) (dual of [2401, 2376, 14]-code), using
- net defined by OOA [i] based on linear OOA(4925, 400, F49, 13, 13) (dual of [(400, 13), 5175, 14]-NRT-code), using
- digital (2, 8, 52)-net over F49, using
(20, 20+13, 4912)-Net over F49 — Digital
Digital (20, 33, 4912)-net over F49, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4933, 4912, F49, 13) (dual of [4912, 4879, 14]-code), using
- 2501 step Varšamov–Edel lengthening with (ri) = (3, 5 times 0, 1, 25 times 0, 1, 96 times 0, 1, 308 times 0, 1, 758 times 0, 1, 1303 times 0) [i] based on linear OA(4925, 2403, F49, 13) (dual of [2403, 2378, 14]-code), using
- construction X applied to Ce(12) ⊂ Ce(11) [i] based on
- linear OA(4925, 2401, F49, 13) (dual of [2401, 2376, 14]-code), using an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 2400 = 492−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- 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(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(12) ⊂ Ce(11) [i] based on
- 2501 step Varšamov–Edel lengthening with (ri) = (3, 5 times 0, 1, 25 times 0, 1, 96 times 0, 1, 308 times 0, 1, 758 times 0, 1, 1303 times 0) [i] based on linear OA(4925, 2403, F49, 13) (dual of [2403, 2378, 14]-code), using
(20, 20+13, large)-Net in Base 49 — Upper bound on s
There is no (20, 33, large)-net in base 49, because
- 11 times m-reduction [i] would yield (20, 22, large)-net in base 49, but