Best Known (67, 67+34, s)-Nets in Base 16
(67, 67+34, 771)-Net over F16 — Constructive and digital
Digital (67, 101, 771)-net over F16, using
- (u, u+v)-construction [i] based on
- digital (16, 33, 257)-net over F16, using
- base reduction for projective spaces (embedding PG(16,256) in PG(32,16)) for nets [i] based on digital (0, 17, 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
- base reduction for projective spaces (embedding PG(16,256) in PG(32,16)) for nets [i] based on digital (0, 17, 257)-net over F256, using
- digital (34, 68, 514)-net over F16, using
- trace code for nets [i] based on digital (0, 34, 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, 34, 257)-net over F256, using
- digital (16, 33, 257)-net over F16, using
(67, 67+34, 4391)-Net over F16 — Digital
Digital (67, 101, 4391)-net over F16, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(16101, 4391, F16, 34) (dual of [4391, 4290, 35]-code), using
- 285 step Varšamov–Edel lengthening with (ri) = (3, 0, 0, 1, 11 times 0, 1, 30 times 0, 1, 76 times 0, 1, 161 times 0) [i] based on linear OA(1694, 4099, F16, 34) (dual of [4099, 4005, 35]-code), using
- construction X applied to Ce(33) ⊂ Ce(32) [i] based on
- linear OA(1694, 4096, F16, 34) (dual of [4096, 4002, 35]-code), using an extension Ce(33) of the primitive narrow-sense BCH-code C(I) with length 4095 = 163−1, defining interval I = [1,33], and designed minimum distance d ≥ |I|+1 = 34 [i]
- linear OA(1691, 4096, F16, 33) (dual of [4096, 4005, 34]-code), using an extension Ce(32) of the primitive narrow-sense BCH-code C(I) with length 4095 = 163−1, defining interval I = [1,32], and designed minimum distance d ≥ |I|+1 = 33 [i]
- linear OA(160, 3, F16, 0) (dual of [3, 3, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(160, s, F16, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(33) ⊂ Ce(32) [i] based on
- 285 step Varšamov–Edel lengthening with (ri) = (3, 0, 0, 1, 11 times 0, 1, 30 times 0, 1, 76 times 0, 1, 161 times 0) [i] based on linear OA(1694, 4099, F16, 34) (dual of [4099, 4005, 35]-code), using
(67, 67+34, 6819345)-Net in Base 16 — Upper bound on s
There is no (67, 101, 6819346)-net in base 16, because
- the generalized Rao bound for nets shows that 16m ≥ 41 316081 276543 464441 359152 467943 881741 531852 138218 674664 684612 018961 401961 217171 306907 800636 666299 530346 209155 397947 976831 > 16101 [i]