Best Known (82−21, 82, s)-Nets in Base 4
(82−21, 82, 514)-Net over F4 — Constructive and digital
Digital (61, 82, 514)-net over F4, using
- trace code for nets [i] based on digital (20, 41, 257)-net over F16, using
- base reduction for projective spaces (embedding PG(20,256) in PG(40,16)) for nets [i] based on digital (0, 21, 257)-net over F256, using
- net from sequence [i] based on digital (0, 256)-sequence over F256, using
- generalized Faure sequence [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F256 with g(F) = 0 and N(F) ≥ 257, using
- the rational function field F256(x) [i]
- Niederreiter sequence [i]
- net from sequence [i] based on digital (0, 256)-sequence over F256, using
- base reduction for projective spaces (embedding PG(20,256) in PG(40,16)) for nets [i] based on digital (0, 21, 257)-net over F256, using
(82−21, 82, 960)-Net over F4 — Digital
Digital (61, 82, 960)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(482, 960, F4, 21) (dual of [960, 878, 22]-code), using
- discarding factors / shortening the dual code based on linear OA(482, 1036, F4, 21) (dual of [1036, 954, 22]-code), using
- construction X applied to C([0,10]) ⊂ C([0,9]) [i] based on
- linear OA(481, 1025, F4, 21) (dual of [1025, 944, 22]-code), using the expurgated narrow-sense BCH-code C(I) with length 1025 | 410−1, defining interval I = [0,10], and minimum distance d ≥ |{−10,−9,…,10}|+1 = 22 (BCH-bound) [i]
- linear OA(471, 1025, F4, 19) (dual of [1025, 954, 20]-code), using the expurgated narrow-sense BCH-code C(I) with length 1025 | 410−1, defining interval I = [0,9], and minimum distance d ≥ |{−9,−8,…,9}|+1 = 20 (BCH-bound) [i]
- linear OA(41, 11, F4, 1) (dual of [11, 10, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(41, s, F4, 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
- discarding factors / shortening the dual code based on linear OA(482, 1036, F4, 21) (dual of [1036, 954, 22]-code), using
(82−21, 82, 113634)-Net in Base 4 — Upper bound on s
There is no (61, 82, 113635)-net in base 4, because
- 1 times m-reduction [i] would yield (61, 81, 113635)-net in base 4, but
- the generalized Rao bound for nets shows that 4m ≥ 5 846151 100303 642939 781576 177757 087376 250085 459220 > 481 [i]