Best Known (31, 58, s)-Nets in Base 32
(31, 58, 218)-Net over F32 — Constructive and digital
Digital (31, 58, 218)-net over F32, using
- (u, u+v)-construction [i] based on
- digital (7, 20, 98)-net over F32, using
- net from sequence [i] based on digital (7, 97)-sequence over F32, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F32 with g(F) = 7 and N(F) ≥ 98, using
- net from sequence [i] based on digital (7, 97)-sequence over F32, using
- digital (11, 38, 120)-net over F32, using
- net from sequence [i] based on digital (11, 119)-sequence over F32, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F32 with g(F) = 11 and N(F) ≥ 120, using
- net from sequence [i] based on digital (11, 119)-sequence over F32, using
- digital (7, 20, 98)-net over F32, using
(31, 58, 290)-Net in Base 32 — Constructive
(31, 58, 290)-net in base 32, using
- (u, u+v)-construction [i] based on
- digital (0, 13, 33)-net over F32, using
- net from sequence [i] based on digital (0, 32)-sequence over F32, using
- generalized Faure sequence [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F32 with g(F) = 0 and N(F) ≥ 33, using
- the rational function field F32(x) [i]
- Niederreiter sequence [i]
- net from sequence [i] based on digital (0, 32)-sequence over F32, using
- (18, 45, 257)-net in base 32, using
- 3 times m-reduction [i] based on (18, 48, 257)-net in base 32, using
- base change [i] based on digital (0, 30, 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 change [i] based on digital (0, 30, 257)-net over F256, using
- 3 times m-reduction [i] based on (18, 48, 257)-net in base 32, using
- digital (0, 13, 33)-net over F32, using
(31, 58, 876)-Net over F32 — Digital
Digital (31, 58, 876)-net over F32, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3258, 876, F32, 27) (dual of [876, 818, 28]-code), using
- discarding factors / shortening the dual code based on linear OA(3258, 1042, F32, 27) (dual of [1042, 984, 28]-code), using
- construction X applied to C([0,13]) ⊂ C([0,10]) [i] based on
- linear OA(3253, 1025, F32, 27) (dual of [1025, 972, 28]-code), using the expurgated narrow-sense BCH-code C(I) with length 1025 | 324−1, defining interval I = [0,13], and minimum distance d ≥ |{−13,−12,…,13}|+1 = 28 (BCH-bound) [i]
- linear OA(3241, 1025, F32, 21) (dual of [1025, 984, 22]-code), using the expurgated narrow-sense BCH-code C(I) with length 1025 | 324−1, defining interval I = [0,10], and minimum distance d ≥ |{−10,−9,…,10}|+1 = 22 (BCH-bound) [i]
- linear OA(325, 17, F32, 5) (dual of [17, 12, 6]-code or 17-arc in PG(4,32)), using
- discarding factors / shortening the dual code based on linear OA(325, 32, F32, 5) (dual of [32, 27, 6]-code or 32-arc in PG(4,32)), using
- Reed–Solomon code RS(27,32) [i]
- discarding factors / shortening the dual code based on linear OA(325, 32, F32, 5) (dual of [32, 27, 6]-code or 32-arc in PG(4,32)), using
- construction X applied to C([0,13]) ⊂ C([0,10]) [i] based on
- discarding factors / shortening the dual code based on linear OA(3258, 1042, F32, 27) (dual of [1042, 984, 28]-code), using
(31, 58, 727016)-Net in Base 32 — Upper bound on s
There is no (31, 58, 727017)-net in base 32, because
- 1 times m-reduction [i] would yield (31, 57, 727017)-net in base 32, but
- the generalized Rao bound for nets shows that 32m ≥ 62 165712 353480 078251 168606 213838 148811 835388 204001 935041 864684 400464 088601 868868 023520 > 3257 [i]