Best Known (20, 20+14, s)-Nets in Base 49
(20, 20+14, 393)-Net over F49 — Constructive and digital
Digital (20, 34, 393)-net over F49, using
- (u, u+v)-construction [i] based on
- digital (0, 7, 50)-net over F49, using
- net from sequence [i] based on digital (0, 49)-sequence over F49, using
- generalized Faure sequence [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F49 with g(F) = 0 and N(F) ≥ 50, using
- the rational function field F49(x) [i]
- Niederreiter sequence [i]
- net from sequence [i] based on digital (0, 49)-sequence over F49, using
- digital (13, 27, 343)-net over F49, using
- net defined by OOA [i] based on linear OOA(4927, 343, F49, 14, 14) (dual of [(343, 14), 4775, 15]-NRT-code), using
- OA 7-folding and stacking [i] based on linear OA(4927, 2401, F49, 14) (dual of [2401, 2374, 15]-code), using
- an extension Ce(13) of the primitive narrow-sense BCH-code C(I) with length 2400 = 492−1, defining interval I = [1,13], and designed minimum distance d ≥ |I|+1 = 14 [i]
- OA 7-folding and stacking [i] based on linear OA(4927, 2401, F49, 14) (dual of [2401, 2374, 15]-code), using
- net defined by OOA [i] based on linear OOA(4927, 343, F49, 14, 14) (dual of [(343, 14), 4775, 15]-NRT-code), using
- digital (0, 7, 50)-net over F49, using
(20, 20+14, 3212)-Net over F49 — Digital
Digital (20, 34, 3212)-net over F49, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4934, 3212, F49, 14) (dual of [3212, 3178, 15]-code), using
- 802 step Varšamov–Edel lengthening with (ri) = (3, 0, 0, 0, 1, 14 times 0, 1, 57 times 0, 1, 194 times 0, 1, 529 times 0) [i] based on linear OA(4927, 2403, F49, 14) (dual of [2403, 2376, 15]-code), using
- construction X applied to Ce(13) ⊂ Ce(12) [i] based on
- linear OA(4927, 2401, F49, 14) (dual of [2401, 2374, 15]-code), using an extension Ce(13) of the primitive narrow-sense BCH-code C(I) with length 2400 = 492−1, defining interval I = [1,13], and designed minimum distance d ≥ |I|+1 = 14 [i]
- 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(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(13) ⊂ Ce(12) [i] based on
- 802 step Varšamov–Edel lengthening with (ri) = (3, 0, 0, 0, 1, 14 times 0, 1, 57 times 0, 1, 194 times 0, 1, 529 times 0) [i] based on linear OA(4927, 2403, F49, 14) (dual of [2403, 2376, 15]-code), using
(20, 20+14, large)-Net in Base 49 — Upper bound on s
There is no (20, 34, large)-net in base 49, because
- 12 times m-reduction [i] would yield (20, 22, large)-net in base 49, but