Best Known (32, 32+22, s)-Nets in Base 49
(32, 32+22, 394)-Net over F49 — Constructive and digital
Digital (32, 54, 394)-net over F49, using
- (u, u+v)-construction [i] based on
- digital (0, 11, 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 (21, 43, 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
- digital (0, 11, 50)-net over F49, using
(32, 32+22, 4025)-Net over F49 — Digital
Digital (32, 54, 4025)-net over F49, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4954, 4025, F49, 22) (dual of [4025, 3971, 23]-code), using
- 1611 step Varšamov–Edel lengthening with (ri) = (4, 0, 1, 6 times 0, 1, 17 times 0, 1, 48 times 0, 1, 121 times 0, 1, 271 times 0, 1, 479 times 0, 1, 660 times 0) [i] based on linear OA(4943, 2403, F49, 22) (dual of [2403, 2360, 23]-code), using
- construction X applied to Ce(21) ⊂ Ce(20) [i] based on
- linear OA(4943, 2401, F49, 22) (dual of [2401, 2358, 23]-code), using an extension Ce(21) of the primitive narrow-sense BCH-code C(I) with length 2400 = 492−1, defining interval I = [1,21], and designed minimum distance d ≥ |I|+1 = 22 [i]
- linear OA(4941, 2401, F49, 21) (dual of [2401, 2360, 22]-code), using an extension Ce(20) of the primitive narrow-sense BCH-code C(I) with length 2400 = 492−1, defining interval I = [1,20], and designed minimum distance d ≥ |I|+1 = 21 [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(21) ⊂ Ce(20) [i] based on
- 1611 step Varšamov–Edel lengthening with (ri) = (4, 0, 1, 6 times 0, 1, 17 times 0, 1, 48 times 0, 1, 121 times 0, 1, 271 times 0, 1, 479 times 0, 1, 660 times 0) [i] based on linear OA(4943, 2403, F49, 22) (dual of [2403, 2360, 23]-code), using
(32, 32+22, large)-Net in Base 49 — Upper bound on s
There is no (32, 54, large)-net in base 49, because
- 20 times m-reduction [i] would yield (32, 34, large)-net in base 49, but