Best Known (248−49, 248, s)-Nets in Base 3
(248−49, 248, 896)-Net over F3 — Constructive and digital
Digital (199, 248, 896)-net over F3, using
- trace code for nets [i] based on digital (13, 62, 224)-net over F81, using
- net from sequence [i] based on digital (13, 223)-sequence over F81, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F81 with g(F) = 13 and N(F) ≥ 224, using
- net from sequence [i] based on digital (13, 223)-sequence over F81, using
(248−49, 248, 2759)-Net over F3 — Digital
Digital (199, 248, 2759)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3248, 2759, F3, 49) (dual of [2759, 2511, 50]-code), using
- 548 step Varšamov–Edel lengthening with (ri) = (4, 1, 1, 0, 1, 0, 1, 0, 0, 0, 1, 5 times 0, 1, 7 times 0, 1, 11 times 0, 1, 15 times 0, 1, 20 times 0, 1, 25 times 0, 1, 32 times 0, 1, 38 times 0, 1, 43 times 0, 1, 47 times 0, 1, 51 times 0, 1, 54 times 0, 1, 57 times 0, 1, 58 times 0, 1, 60 times 0) [i] based on linear OA(3225, 2188, F3, 49) (dual of [2188, 1963, 50]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 2188 | 314−1, defining interval I = [0,24], and minimum distance d ≥ |{−24,−23,…,24}|+1 = 50 (BCH-bound) [i]
- 548 step Varšamov–Edel lengthening with (ri) = (4, 1, 1, 0, 1, 0, 1, 0, 0, 0, 1, 5 times 0, 1, 7 times 0, 1, 11 times 0, 1, 15 times 0, 1, 20 times 0, 1, 25 times 0, 1, 32 times 0, 1, 38 times 0, 1, 43 times 0, 1, 47 times 0, 1, 51 times 0, 1, 54 times 0, 1, 57 times 0, 1, 58 times 0, 1, 60 times 0) [i] based on linear OA(3225, 2188, F3, 49) (dual of [2188, 1963, 50]-code), using
(248−49, 248, 398731)-Net in Base 3 — Upper bound on s
There is no (199, 248, 398732)-net in base 3, because
- 1 times m-reduction [i] would yield (199, 247, 398732)-net in base 3, but
- the generalized Rao bound for nets shows that 3m ≥ 7062 763788 168778 556996 595639 742991 761697 369228 778944 377155 725599 736164 962958 378637 242288 072328 748539 872505 994337 467969 > 3247 [i]