Best Known (64, 64+33, s)-Nets in Base 16
(64, 64+33, 771)-Net over F16 — Constructive and digital
Digital (64, 97, 771)-net over F16, using
- (u, u+v)-construction [i] based on
- digital (15, 31, 257)-net over F16, using
- base reduction for projective spaces (embedding PG(15,256) in PG(30,16)) for nets [i] based on digital (0, 16, 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(15,256) in PG(30,16)) for nets [i] based on digital (0, 16, 257)-net over F256, using
- digital (33, 66, 514)-net over F16, using
- trace code for nets [i] based on digital (0, 33, 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, 33, 257)-net over F256, using
- digital (15, 31, 257)-net over F16, using
(64, 64+33, 4203)-Net over F16 — Digital
Digital (64, 97, 4203)-net over F16, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(1697, 4203, F16, 33) (dual of [4203, 4106, 34]-code), using
- 100 step Varšamov–Edel lengthening with (ri) = (3, 0, 0, 1, 9 times 0, 1, 24 times 0, 1, 61 times 0) [i] based on linear OA(1691, 4097, F16, 33) (dual of [4097, 4006, 34]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 4097 | 166−1, defining interval I = [0,16], and minimum distance d ≥ |{−16,−15,…,16}|+1 = 34 (BCH-bound) [i]
- 100 step Varšamov–Edel lengthening with (ri) = (3, 0, 0, 1, 9 times 0, 1, 24 times 0, 1, 61 times 0) [i] based on linear OA(1691, 4097, F16, 33) (dual of [4097, 4006, 34]-code), using
(64, 64+33, 7606184)-Net in Base 16 — Upper bound on s
There is no (64, 97, 7606185)-net in base 16, because
- 1 times m-reduction [i] would yield (64, 96, 7606185)-net in base 16, but
- the generalized Rao bound for nets shows that 16m ≥ 39 402023 461684 377686 922446 588952 510028 832744 090129 487434 205908 335062 896316 421149 298604 200285 910505 697563 543247 610026 > 1696 [i]