Best Known (115−24, 115, s)-Nets in Base 4
(115−24, 115, 1049)-Net over F4 — Constructive and digital
Digital (91, 115, 1049)-net over F4, using
- (u, u+v)-construction [i] based on
- digital (7, 19, 21)-net over F4, using
- net from sequence [i] based on digital (7, 20)-sequence over F4, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F4 with g(F) = 7 and N(F) ≥ 21, using
- net from sequence [i] based on digital (7, 20)-sequence over F4, using
- digital (72, 96, 1028)-net over F4, using
- trace code for nets [i] based on digital (0, 24, 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, 24, 257)-net over F256, using
- digital (7, 19, 21)-net over F4, using
(115−24, 115, 3959)-Net over F4 — Digital
Digital (91, 115, 3959)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4115, 3959, F4, 24) (dual of [3959, 3844, 25]-code), using
- discarding factors / shortening the dual code based on linear OA(4115, 4118, F4, 24) (dual of [4118, 4003, 25]-code), using
- construction X applied to C([0,12]) ⊂ C([0,9]) [i] based on
- linear OA(4109, 4097, F4, 25) (dual of [4097, 3988, 26]-code), using the expurgated narrow-sense BCH-code C(I) with length 4097 | 412−1, defining interval I = [0,12], and minimum distance d ≥ |{−12,−11,…,12}|+1 = 26 (BCH-bound) [i]
- linear OA(485, 4097, F4, 19) (dual of [4097, 4012, 20]-code), using the expurgated narrow-sense BCH-code C(I) with length 4097 | 412−1, defining interval I = [0,9], and minimum distance d ≥ |{−9,−8,…,9}|+1 = 20 (BCH-bound) [i]
- linear OA(46, 21, F4, 4) (dual of [21, 15, 5]-code), using
- construction X applied to C([0,12]) ⊂ C([0,9]) [i] based on
- discarding factors / shortening the dual code based on linear OA(4115, 4118, F4, 24) (dual of [4118, 4003, 25]-code), using
(115−24, 115, 1037475)-Net in Base 4 — Upper bound on s
There is no (91, 115, 1037476)-net in base 4, because
- the generalized Rao bound for nets shows that 4m ≥ 1725 450563 141046 002401 692398 201121 585912 273588 221690 280292 617083 096056 > 4115 [i]