Best Known (11, 19, s)-Nets in Base 49
(11, 19, 650)-Net over F49 — Constructive and digital
Digital (11, 19, 650)-net over F49, using
- (u, u+v)-construction [i] based on
- digital (0, 4, 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 (7, 15, 600)-net over F49, using
- net defined by OOA [i] based on linear OOA(4915, 600, F49, 8, 8) (dual of [(600, 8), 4785, 9]-NRT-code), using
- OA 4-folding and stacking [i] based on linear OA(4915, 2400, F49, 8) (dual of [2400, 2385, 9]-code), using
- discarding factors / shortening the dual code based on linear OA(4915, 2401, F49, 8) (dual of [2401, 2386, 9]-code), using
- an extension Ce(7) of the primitive narrow-sense BCH-code C(I) with length 2400 = 492−1, defining interval I = [1,7], and designed minimum distance d ≥ |I|+1 = 8 [i]
- discarding factors / shortening the dual code based on linear OA(4915, 2401, F49, 8) (dual of [2401, 2386, 9]-code), using
- OA 4-folding and stacking [i] based on linear OA(4915, 2400, F49, 8) (dual of [2400, 2385, 9]-code), using
- net defined by OOA [i] based on linear OOA(4915, 600, F49, 8, 8) (dual of [(600, 8), 4785, 9]-NRT-code), using
- digital (0, 4, 50)-net over F49, using
(11, 19, 3050)-Net over F49 — Digital
Digital (11, 19, 3050)-net over F49, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4919, 3050, F49, 8) (dual of [3050, 3031, 9]-code), using
- 643 step Varšamov–Edel lengthening with (ri) = (2, 12 times 0, 1, 94 times 0, 1, 534 times 0) [i] based on linear OA(4915, 2403, F49, 8) (dual of [2403, 2388, 9]-code), using
- construction X applied to Ce(7) ⊂ Ce(6) [i] based on
- linear OA(4915, 2401, F49, 8) (dual of [2401, 2386, 9]-code), using an extension Ce(7) of the primitive narrow-sense BCH-code C(I) with length 2400 = 492−1, defining interval I = [1,7], and designed minimum distance d ≥ |I|+1 = 8 [i]
- linear OA(4913, 2401, F49, 7) (dual of [2401, 2388, 8]-code), using an extension Ce(6) of the primitive narrow-sense BCH-code C(I) with length 2400 = 492−1, defining interval I = [1,6], and designed minimum distance d ≥ |I|+1 = 7 [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(7) ⊂ Ce(6) [i] based on
- 643 step Varšamov–Edel lengthening with (ri) = (2, 12 times 0, 1, 94 times 0, 1, 534 times 0) [i] based on linear OA(4915, 2403, F49, 8) (dual of [2403, 2388, 9]-code), using
(11, 19, 4923147)-Net in Base 49 — Upper bound on s
There is no (11, 19, 4923148)-net in base 49, because
- the generalized Rao bound for nets shows that 49m ≥ 129 934905 085185 361116 265902 685441 > 4919 [i]