Best Known (29, 29+20, s)-Nets in Base 49
(29, 29+20, 344)-Net over F49 — Constructive and digital
Digital (29, 49, 344)-net over F49, using
- t-expansion [i] based on digital (21, 49, 344)-net over F49, using
- net from sequence [i] based on digital (21, 343)-sequence over F49, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F49 with g(F) = 21 and N(F) ≥ 344, using
- the Hermitian function field over F49 [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F49 with g(F) = 21 and N(F) ≥ 344, using
- net from sequence [i] based on digital (21, 343)-sequence over F49, using
(29, 29+20, 3793)-Net over F49 — Digital
Digital (29, 49, 3793)-net over F49, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4949, 3793, F49, 20) (dual of [3793, 3744, 21]-code), using
- 1380 step Varšamov–Edel lengthening with (ri) = (4, 0, 0, 0, 1, 10 times 0, 1, 30 times 0, 1, 82 times 0, 1, 197 times 0, 1, 405 times 0, 1, 646 times 0) [i] based on linear OA(4939, 2403, F49, 20) (dual of [2403, 2364, 21]-code), using
- construction X applied to Ce(19) ⊂ Ce(18) [i] based on
- linear OA(4939, 2401, F49, 20) (dual of [2401, 2362, 21]-code), using an extension Ce(19) of the primitive narrow-sense BCH-code C(I) with length 2400 = 492−1, defining interval I = [1,19], and designed minimum distance d ≥ |I|+1 = 20 [i]
- linear OA(4937, 2401, F49, 19) (dual of [2401, 2364, 20]-code), using an extension Ce(18) of the primitive narrow-sense BCH-code C(I) with length 2400 = 492−1, defining interval I = [1,18], and designed minimum distance d ≥ |I|+1 = 19 [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(19) ⊂ Ce(18) [i] based on
- 1380 step Varšamov–Edel lengthening with (ri) = (4, 0, 0, 0, 1, 10 times 0, 1, 30 times 0, 1, 82 times 0, 1, 197 times 0, 1, 405 times 0, 1, 646 times 0) [i] based on linear OA(4939, 2403, F49, 20) (dual of [2403, 2364, 21]-code), using
(29, 29+20, large)-Net in Base 49 — Upper bound on s
There is no (29, 49, large)-net in base 49, because
- 18 times m-reduction [i] would yield (29, 31, large)-net in base 49, but