Best Known (80, 80+21, s)-Nets in Base 3
(80, 80+21, 600)-Net over F3 — Constructive and digital
Digital (80, 101, 600)-net over F3, using
- 31 times duplication [i] based on digital (79, 100, 600)-net over F3, using
- trace code for nets [i] based on digital (4, 25, 150)-net over F81, using
- net from sequence [i] based on digital (4, 149)-sequence over F81, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F81 with g(F) = 4 and N(F) ≥ 150, using
- net from sequence [i] based on digital (4, 149)-sequence over F81, using
- trace code for nets [i] based on digital (4, 25, 150)-net over F81, using
(80, 80+21, 1269)-Net over F3 — Digital
Digital (80, 101, 1269)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3101, 1269, F3, 21) (dual of [1269, 1168, 22]-code), using
- discarding factors / shortening the dual code based on linear OA(3101, 2204, F3, 21) (dual of [2204, 2103, 22]-code), using
- 1 times code embedding in larger space [i] based on linear OA(3100, 2203, F3, 21) (dual of [2203, 2103, 22]-code), using
- construction X applied to C([0,10]) ⊂ C([0,9]) [i] based on
- linear OA(399, 2188, F3, 21) (dual of [2188, 2089, 22]-code), using the expurgated narrow-sense BCH-code C(I) with length 2188 | 314−1, defining interval I = [0,10], and minimum distance d ≥ |{−10,−9,…,10}|+1 = 22 (BCH-bound) [i]
- linear OA(385, 2188, F3, 19) (dual of [2188, 2103, 20]-code), using the expurgated narrow-sense BCH-code C(I) with length 2188 | 314−1, defining interval I = [0,9], and minimum distance d ≥ |{−9,−8,…,9}|+1 = 20 (BCH-bound) [i]
- linear OA(31, 15, F3, 1) (dual of [15, 14, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(31, s, F3, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to C([0,10]) ⊂ C([0,9]) [i] based on
- 1 times code embedding in larger space [i] based on linear OA(3100, 2203, F3, 21) (dual of [2203, 2103, 22]-code), using
- discarding factors / shortening the dual code based on linear OA(3101, 2204, F3, 21) (dual of [2204, 2103, 22]-code), using
(80, 80+21, 133698)-Net in Base 3 — Upper bound on s
There is no (80, 101, 133699)-net in base 3, because
- 1 times m-reduction [i] would yield (80, 100, 133699)-net in base 3, but
- the generalized Rao bound for nets shows that 3m ≥ 515379 431190 280470 307568 433529 389619 174810 046421 > 3100 [i]