Best Known (39, 39+19, s)-Nets in Base 4
(39, 39+19, 195)-Net over F4 — Constructive and digital
Digital (39, 58, 195)-net over F4, using
- 41 times duplication [i] based on digital (38, 57, 195)-net over F4, using
- trace code for nets [i] based on digital (0, 19, 65)-net over F64, using
- net from sequence [i] based on digital (0, 64)-sequence over F64, using
- generalized Faure sequence [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F64 with g(F) = 0 and N(F) ≥ 65, using
- the rational function field F64(x) [i]
- Niederreiter sequence [i]
- net from sequence [i] based on digital (0, 64)-sequence over F64, using
- trace code for nets [i] based on digital (0, 19, 65)-net over F64, using
(39, 39+19, 238)-Net over F4 — Digital
Digital (39, 58, 238)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(458, 238, F4, 19) (dual of [238, 180, 20]-code), using
- discarding factors / shortening the dual code based on linear OA(458, 266, F4, 19) (dual of [266, 208, 20]-code), using
- construction X applied to C([0,9]) ⊂ C([0,8]) [i] based on
- linear OA(457, 257, F4, 19) (dual of [257, 200, 20]-code), using the expurgated narrow-sense BCH-code C(I) with length 257 | 48−1, defining interval I = [0,9], and minimum distance d ≥ |{−9,−8,…,9}|+1 = 20 (BCH-bound) [i]
- linear OA(449, 257, F4, 17) (dual of [257, 208, 18]-code), using the expurgated narrow-sense BCH-code C(I) with length 257 | 48−1, defining interval I = [0,8], and minimum distance d ≥ |{−8,−7,…,8}|+1 = 18 (BCH-bound) [i]
- linear OA(41, 9, F4, 1) (dual of [9, 8, 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([0,9]) ⊂ C([0,8]) [i] based on
- discarding factors / shortening the dual code based on linear OA(458, 266, F4, 19) (dual of [266, 208, 20]-code), using
(39, 39+19, 8981)-Net in Base 4 — Upper bound on s
There is no (39, 58, 8982)-net in base 4, because
- 1 times m-reduction [i] would yield (39, 57, 8982)-net in base 4, but
- the generalized Rao bound for nets shows that 4m ≥ 20784 029424 372182 769869 195317 255078 > 457 [i]