Best Known (39, 39+17, s)-Nets in Base 4
(39, 39+17, 240)-Net over F4 — Constructive and digital
Digital (39, 56, 240)-net over F4, using
- 1 times m-reduction [i] based on digital (39, 57, 240)-net over F4, using
- trace code for nets [i] based on digital (1, 19, 80)-net over F64, using
- net from sequence [i] based on digital (1, 79)-sequence over F64, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F64 with g(F) = 1 and N(F) ≥ 80, using
- net from sequence [i] based on digital (1, 79)-sequence over F64, using
- trace code for nets [i] based on digital (1, 19, 80)-net over F64, using
(39, 39+17, 304)-Net over F4 — Digital
Digital (39, 56, 304)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(456, 304, F4, 17) (dual of [304, 248, 18]-code), using
- 40 step Varšamov–Edel lengthening with (ri) = (2, 1, 1, 0, 0, 1, 6 times 0, 1, 10 times 0, 1, 16 times 0) [i] based on linear OA(449, 257, F4, 17) (dual of [257, 208, 18]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 257 | 48−1, defining interval I = [0,8], and minimum distance d ≥ |{−8,−7,…,8}|+1 = 18 (BCH-bound) [i]
- 40 step Varšamov–Edel lengthening with (ri) = (2, 1, 1, 0, 0, 1, 6 times 0, 1, 10 times 0, 1, 16 times 0) [i] based on linear OA(449, 257, F4, 17) (dual of [257, 208, 18]-code), using
(39, 39+17, 17281)-Net in Base 4 — Upper bound on s
There is no (39, 56, 17282)-net in base 4, because
- 1 times m-reduction [i] would yield (39, 55, 17282)-net in base 4, but
- the generalized Rao bound for nets shows that 4m ≥ 1298 496743 746261 188867 271975 050256 > 455 [i]