Best Known (210, 210+38, s)-Nets in Base 3
(210, 210+38, 1504)-Net over F3 — Constructive and digital
Digital (210, 248, 1504)-net over F3, using
- (u, u+v)-construction [i] based on
- digital (13, 32, 24)-net over F3, using
- net from sequence [i] based on digital (13, 23)-sequence over F3, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F3 with g(F) = 13 and N(F) ≥ 24, using
- net from sequence [i] based on digital (13, 23)-sequence over F3, using
- digital (178, 216, 1480)-net over F3, using
- trace code for nets [i] based on digital (16, 54, 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, 54, 370)-net over F81, using
- digital (13, 32, 24)-net over F3, using
(210, 210+38, 13371)-Net over F3 — Digital
Digital (210, 248, 13371)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3248, 13371, F3, 38) (dual of [13371, 13123, 39]-code), using
- discarding factors / shortening the dual code based on linear OA(3248, 19759, F3, 38) (dual of [19759, 19511, 39]-code), using
- 1 times code embedding in larger space [i] based on linear OA(3247, 19758, F3, 38) (dual of [19758, 19511, 39]-code), using
- construction X applied to Ce(37) ⊂ Ce(28) [i] based on
- linear OA(3226, 19683, F3, 38) (dual of [19683, 19457, 39]-code), using an extension Ce(37) of the primitive narrow-sense BCH-code C(I) with length 19682 = 39−1, defining interval I = [1,37], and designed minimum distance d ≥ |I|+1 = 38 [i]
- linear OA(3172, 19683, F3, 29) (dual of [19683, 19511, 30]-code), using an extension Ce(28) of the primitive narrow-sense BCH-code C(I) with length 19682 = 39−1, defining interval I = [1,28], and designed minimum distance d ≥ |I|+1 = 29 [i]
- linear OA(321, 75, F3, 8) (dual of [75, 54, 9]-code), using
- discarding factors / shortening the dual code based on linear OA(321, 80, F3, 8) (dual of [80, 59, 9]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 80 = 34−1, defining interval I = [0,7], and designed minimum distance d ≥ |I|+1 = 9 [i]
- discarding factors / shortening the dual code based on linear OA(321, 80, F3, 8) (dual of [80, 59, 9]-code), using
- construction X applied to Ce(37) ⊂ Ce(28) [i] based on
- 1 times code embedding in larger space [i] based on linear OA(3247, 19758, F3, 38) (dual of [19758, 19511, 39]-code), using
- discarding factors / shortening the dual code based on linear OA(3248, 19759, F3, 38) (dual of [19759, 19511, 39]-code), using
(210, 210+38, 6696876)-Net in Base 3 — Upper bound on s
There is no (210, 248, 6696877)-net in base 3, because
- the generalized Rao bound for nets shows that 3m ≥ 21187 134191 206121 047637 172017 129252 643184 425677 702414 480210 188027 967263 592000 410141 249712 823585 863519 599895 105491 516299 > 3248 [i]