Best Known (249−40, 249, s)-Nets in Base 3
(249−40, 249, 1493)-Net over F3 — Constructive and digital
Digital (209, 249, 1493)-net over F3, using
- (u, u+v)-construction [i] based on
- digital (5, 25, 13)-net over F3, using
- net from sequence [i] based on digital (5, 12)-sequence over F3, using
- Niederreiter–Xing sequence construction III based on the algebraic function field F/F3 with g(F) = 4, N(F) = 12, and 1 place with degree 2 [i] based on function field F/F3 with g(F) = 4 and N(F) ≥ 12, using an explicitly constructive algebraic function field [i]
- net from sequence [i] based on digital (5, 12)-sequence over F3, using
- digital (184, 224, 1480)-net over F3, using
- trace code for nets [i] based on digital (16, 56, 370)-net over F81, using
- net from sequence [i] based on digital (16, 369)-sequence over F81, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F81 with g(F) = 16 and N(F) ≥ 370, using
- net from sequence [i] based on digital (16, 369)-sequence over F81, using
- trace code for nets [i] based on digital (16, 56, 370)-net over F81, using
- digital (5, 25, 13)-net over F3, using
(249−40, 249, 9866)-Net over F3 — Digital
Digital (209, 249, 9866)-net over F3, using
- 31 times duplication [i] based on digital (208, 248, 9866)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3248, 9866, F3, 2, 40) (dual of [(9866, 2), 19484, 41]-NRT-code), using
- 1 times NRT-code embedding in larger space [i] based on linear OOA(3246, 9865, F3, 2, 40) (dual of [(9865, 2), 19484, 41]-NRT-code), using
- OOA 2-folding [i] based on linear OA(3246, 19730, F3, 40) (dual of [19730, 19484, 41]-code), using
- construction X applied to Ce(39) ⊂ Ce(33) [i] based on
- linear OA(3235, 19683, F3, 40) (dual of [19683, 19448, 41]-code), using an extension Ce(39) of the primitive narrow-sense BCH-code C(I) with length 19682 = 39−1, defining interval I = [1,39], and designed minimum distance d ≥ |I|+1 = 40 [i]
- linear OA(3199, 19683, F3, 34) (dual of [19683, 19484, 35]-code), using an extension Ce(33) of the primitive narrow-sense BCH-code C(I) with length 19682 = 39−1, defining interval I = [1,33], and designed minimum distance d ≥ |I|+1 = 34 [i]
- linear OA(311, 47, F3, 5) (dual of [47, 36, 6]-code), using
- discarding factors / shortening the dual code based on linear OA(311, 85, F3, 5) (dual of [85, 74, 6]-code), using
- construction X applied to Ce(39) ⊂ Ce(33) [i] based on
- OOA 2-folding [i] based on linear OA(3246, 19730, F3, 40) (dual of [19730, 19484, 41]-code), using
- 1 times NRT-code embedding in larger space [i] based on linear OOA(3246, 9865, F3, 2, 40) (dual of [(9865, 2), 19484, 41]-NRT-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3248, 9866, F3, 2, 40) (dual of [(9866, 2), 19484, 41]-NRT-code), using
(249−40, 249, 3617708)-Net in Base 3 — Upper bound on s
There is no (209, 249, 3617709)-net in base 3, because
- the generalized Rao bound for nets shows that 3m ≥ 63561 585299 954412 433075 766214 868333 997913 626373 451261 904662 584098 954365 936117 142146 275026 010156 482395 311753 392735 152977 > 3249 [i]