Best Known (56−27, 56, s)-Nets in Base 32
(56−27, 56, 202)-Net over F32 — Constructive and digital
Digital (29, 56, 202)-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 (9, 36, 104)-net over F32, using
- net from sequence [i] based on digital (9, 103)-sequence over F32, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F32 with g(F) = 9 and N(F) ≥ 104, using
- net from sequence [i] based on digital (9, 103)-sequence over F32, using
- digital (7, 20, 98)-net over F32, using
(56−27, 56, 288)-Net in Base 32 — Constructive
(29, 56, 288)-net in base 32, using
- 14 times m-reduction [i] based on (29, 70, 288)-net in base 32, using
- base change [i] based on digital (9, 50, 288)-net over F128, using
- net from sequence [i] based on digital (9, 287)-sequence over F128, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F128 with g(F) = 9 and N(F) ≥ 288, using
- net from sequence [i] based on digital (9, 287)-sequence over F128, using
- base change [i] based on digital (9, 50, 288)-net over F128, using
(56−27, 56, 661)-Net over F32 — Digital
Digital (29, 56, 661)-net over F32, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3256, 661, F32, 27) (dual of [661, 605, 28]-code), using
- discarding factors / shortening the dual code based on linear OA(3256, 1036, F32, 27) (dual of [1036, 980, 28]-code), using
- construction X applied to C([0,13]) ⊂ C([0,11]) [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(3245, 1025, F32, 23) (dual of [1025, 980, 24]-code), using the expurgated narrow-sense BCH-code C(I) with length 1025 | 324−1, defining interval I = [0,11], and minimum distance d ≥ |{−11,−10,…,11}|+1 = 24 (BCH-bound) [i]
- linear OA(323, 11, F32, 3) (dual of [11, 8, 4]-code or 11-arc in PG(2,32) or 11-cap in PG(2,32)), using
- discarding factors / shortening the dual code based on linear OA(323, 32, F32, 3) (dual of [32, 29, 4]-code or 32-arc in PG(2,32) or 32-cap in PG(2,32)), using
- Reed–Solomon code RS(29,32) [i]
- discarding factors / shortening the dual code based on linear OA(323, 32, F32, 3) (dual of [32, 29, 4]-code or 32-arc in PG(2,32) or 32-cap in PG(2,32)), using
- construction X applied to C([0,13]) ⊂ C([0,11]) [i] based on
- discarding factors / shortening the dual code based on linear OA(3256, 1036, F32, 27) (dual of [1036, 980, 28]-code), using
(56−27, 56, 426560)-Net in Base 32 — Upper bound on s
There is no (29, 56, 426561)-net in base 32, because
- 1 times m-reduction [i] would yield (29, 55, 426561)-net in base 32, but
- the generalized Rao bound for nets shows that 32m ≥ 60710 185985 981152 595235 690162 394443 954627 711221 013124 175440 667859 502378 240819 679424 > 3255 [i]