Best Known (217−38, 217, s)-Nets in Base 3
(217−38, 217, 1480)-Net over F3 — Constructive and digital
Digital (179, 217, 1480)-net over F3, using
- 31 times duplication [i] based on 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
(217−38, 217, 5171)-Net over F3 — Digital
Digital (179, 217, 5171)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3217, 5171, F3, 38) (dual of [5171, 4954, 39]-code), using
- discarding factors / shortening the dual code based on linear OA(3217, 6602, F3, 38) (dual of [6602, 6385, 39]-code), using
- construction X applied to C([0,19]) ⊂ C([0,16]) [i] based on
- linear OA(3209, 6562, F3, 39) (dual of [6562, 6353, 40]-code), using the expurgated narrow-sense BCH-code C(I) with length 6562 | 316−1, defining interval I = [0,19], and minimum distance d ≥ |{−19,−18,…,19}|+1 = 40 (BCH-bound) [i]
- linear OA(3177, 6562, F3, 33) (dual of [6562, 6385, 34]-code), using the expurgated narrow-sense BCH-code C(I) with length 6562 | 316−1, defining interval I = [0,16], and minimum distance d ≥ |{−16,−15,…,16}|+1 = 34 (BCH-bound) [i]
- linear OA(38, 40, F3, 4) (dual of [40, 32, 5]-code), using
- discarding factors / shortening the dual code based on linear OA(38, 41, F3, 4) (dual of [41, 33, 5]-code), using
- the narrow-sense BCH-code C(I) with length 41 | 38−1, defining interval I = [1,1], and minimum distance d ≥ |{−3,−1,1,3}|+1 = 5 (BCH-bound) [i]
- discarding factors / shortening the dual code based on linear OA(38, 41, F3, 4) (dual of [41, 33, 5]-code), using
- construction X applied to C([0,19]) ⊂ C([0,16]) [i] based on
- discarding factors / shortening the dual code based on linear OA(3217, 6602, F3, 38) (dual of [6602, 6385, 39]-code), using
(217−38, 217, 1115334)-Net in Base 3 — Upper bound on s
There is no (179, 217, 1115335)-net in base 3, because
- the generalized Rao bound for nets shows that 3m ≥ 34 301526 674397 850688 968504 467897 398175 328868 078096 272014 656493 420245 673584 800732 357180 707197 055200 139771 > 3217 [i]