Best Known (40, 40+11, s)-Nets in Base 4
(40, 40+11, 1044)-Net over F4 — Constructive and digital
Digital (40, 51, 1044)-net over F4, using
- (u, u+v)-construction [i] based on
- digital (2, 7, 16)-net over F4, using
- digital (33, 44, 1028)-net over F4, using
- trace code for nets [i] based on digital (0, 11, 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
- trace code for nets [i] based on digital (0, 11, 257)-net over F256, using
(40, 40+11, 3051)-Net over F4 — Digital
Digital (40, 51, 3051)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(451, 3051, F4, 11) (dual of [3051, 3000, 12]-code), using
- discarding factors / shortening the dual code based on linear OA(451, 4112, F4, 11) (dual of [4112, 4061, 12]-code), using
- 1 times code embedding in larger space [i] based on linear OA(450, 4111, F4, 11) (dual of [4111, 4061, 12]-code), using
- construction X4 applied to C([0,5]) ⊂ C([0,4]) [i] based on
- linear OA(449, 4097, F4, 11) (dual of [4097, 4048, 12]-code), using the expurgated narrow-sense BCH-code C(I) with length 4097 | 412−1, defining interval I = [0,5], and minimum distance d ≥ |{−5,−4,…,5}|+1 = 12 (BCH-bound) [i]
- linear OA(437, 4097, F4, 9) (dual of [4097, 4060, 10]-code), using the expurgated narrow-sense BCH-code C(I) with length 4097 | 412−1, defining interval I = [0,4], and minimum distance d ≥ |{−4,−3,…,4}|+1 = 10 (BCH-bound) [i]
- linear OA(413, 14, F4, 13) (dual of [14, 1, 14]-code or 14-arc in PG(12,4)), using
- dual of repetition code with length 14 [i]
- linear OA(41, 14, F4, 1) (dual of [14, 13, 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 X4 applied to C([0,5]) ⊂ C([0,4]) [i] based on
- 1 times code embedding in larger space [i] based on linear OA(450, 4111, F4, 11) (dual of [4111, 4061, 12]-code), using
- discarding factors / shortening the dual code based on linear OA(451, 4112, F4, 11) (dual of [4112, 4061, 12]-code), using
(40, 40+11, 910569)-Net in Base 4 — Upper bound on s
There is no (40, 51, 910570)-net in base 4, because
- 1 times m-reduction [i] would yield (40, 50, 910570)-net in base 4, but
- the generalized Rao bound for nets shows that 4m ≥ 1 267653 209748 025283 809663 915978 > 450 [i]