Best Known (49, 74, s)-Nets in Base 16
(49, 74, 1028)-Net over F16 — Constructive and digital
Digital (49, 74, 1028)-net over F16, using
- (u, u+v)-construction [i] based on
- digital (12, 24, 514)-net over F16, using
- trace code for nets [i] based on digital (0, 12, 257)-net over F256, using
- net from sequence [i] based on digital (0, 256)-sequence over F256, using
- generalized Faure sequence [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F256 with g(F) = 0 and N(F) ≥ 257, using
- the rational function field F256(x) [i]
- Niederreiter sequence [i]
- net from sequence [i] based on digital (0, 256)-sequence over F256, using
- trace code for nets [i] based on digital (0, 12, 257)-net over F256, using
- digital (25, 50, 514)-net over F16, using
- trace code for nets [i] based on digital (0, 25, 257)-net over F256, using
- net from sequence [i] based on digital (0, 256)-sequence over F256 (see above)
- trace code for nets [i] based on digital (0, 25, 257)-net over F256, using
- digital (12, 24, 514)-net over F16, using
(49, 74, 4127)-Net over F16 — Digital
Digital (49, 74, 4127)-net over F16, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(1674, 4127, F16, 25) (dual of [4127, 4053, 26]-code), using
- 21 step Varšamov–Edel lengthening with (ri) = (2, 0, 0, 0, 1, 16 times 0) [i] based on linear OA(1671, 4103, F16, 25) (dual of [4103, 4032, 26]-code), using
- construction X applied to Ce(24) ⊂ Ce(22) [i] based on
- linear OA(1670, 4096, F16, 25) (dual of [4096, 4026, 26]-code), using an extension Ce(24) of the primitive narrow-sense BCH-code C(I) with length 4095 = 163−1, defining interval I = [1,24], and designed minimum distance d ≥ |I|+1 = 25 [i]
- linear OA(1664, 4096, F16, 23) (dual of [4096, 4032, 24]-code), using an extension Ce(22) of the primitive narrow-sense BCH-code C(I) with length 4095 = 163−1, defining interval I = [1,22], and designed minimum distance d ≥ |I|+1 = 23 [i]
- linear OA(161, 7, F16, 1) (dual of [7, 6, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(161, s, F16, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to Ce(24) ⊂ Ce(22) [i] based on
- 21 step Varšamov–Edel lengthening with (ri) = (2, 0, 0, 0, 1, 16 times 0) [i] based on linear OA(1671, 4103, F16, 25) (dual of [4103, 4032, 26]-code), using
(49, 74, 7453032)-Net in Base 16 — Upper bound on s
There is no (49, 74, 7453033)-net in base 16, because
- 1 times m-reduction [i] would yield (49, 73, 7453033)-net in base 16, but
- the generalized Rao bound for nets shows that 16m ≥ 7957 178122 757073 158889 022857 265959 717553 193215 568766 672002 835277 863524 607108 808572 767416 > 1673 [i]