Best Known (18, 18+13, s)-Nets in Base 49
(18, 18+13, 450)-Net over F49 — Constructive and digital
Digital (18, 31, 450)-net over F49, using
- (u, u+v)-construction [i] based on
- digital (0, 6, 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 (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 (0, 6, 50)-net over F49, using
(18, 18+13, 2847)-Net over F49 — Digital
Digital (18, 31, 2847)-net over F49, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4931, 2847, F49, 13) (dual of [2847, 2816, 14]-code), using
- 438 step Varšamov–Edel lengthening with (ri) = (3, 5 times 0, 1, 25 times 0, 1, 96 times 0, 1, 308 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
- 438 step Varšamov–Edel lengthening with (ri) = (3, 5 times 0, 1, 25 times 0, 1, 96 times 0, 1, 308 times 0) [i] based on linear OA(4925, 2403, F49, 13) (dual of [2403, 2378, 14]-code), using
(18, 18+13, large)-Net in Base 49 — Upper bound on s
There is no (18, 31, large)-net in base 49, because
- 11 times m-reduction [i] would yield (18, 20, large)-net in base 49, but