Best Known (32, 32+23, s)-Nets in Base 49
(32, 32+23, 394)-Net over F49 — Constructive and digital
Digital (32, 55, 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, 44, 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+23, 3207)-Net over F49 — Digital
Digital (32, 55, 3207)-net over F49, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4955, 3207, F49, 23) (dual of [3207, 3152, 24]-code), using
- 794 step Varšamov–Edel lengthening with (ri) = (4, 0, 1, 0, 0, 0, 1, 13 times 0, 1, 38 times 0, 1, 96 times 0, 1, 222 times 0, 1, 414 times 0) [i] based on linear OA(4945, 2403, F49, 23) (dual of [2403, 2358, 24]-code), using
- construction X applied to Ce(22) ⊂ Ce(21) [i] based on
- linear OA(4945, 2401, F49, 23) (dual of [2401, 2356, 24]-code), using an extension Ce(22) of the primitive narrow-sense BCH-code C(I) with length 2400 = 492−1, defining interval I = [1,22], and designed minimum distance d ≥ |I|+1 = 23 [i]
- 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(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(22) ⊂ Ce(21) [i] based on
- 794 step Varšamov–Edel lengthening with (ri) = (4, 0, 1, 0, 0, 0, 1, 13 times 0, 1, 38 times 0, 1, 96 times 0, 1, 222 times 0, 1, 414 times 0) [i] based on linear OA(4945, 2403, F49, 23) (dual of [2403, 2358, 24]-code), using
(32, 32+23, large)-Net in Base 49 — Upper bound on s
There is no (32, 55, large)-net in base 49, because
- 21 times m-reduction [i] would yield (32, 34, large)-net in base 49, but