Best Known (42−21, 42, s)-Nets in Base 32
(42−21, 42, 162)-Net over F32 — Constructive and digital
Digital (21, 42, 162)-net over F32, using
- 1 times m-reduction [i] based on digital (21, 43, 162)-net over F32, using
- (u, u+v)-construction [i] based on
- digital (3, 14, 64)-net over F32, using
- net from sequence [i] based on digital (3, 63)-sequence over F32, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F32 with g(F) = 3 and N(F) ≥ 64, using
- net from sequence [i] based on digital (3, 63)-sequence over F32, using
- digital (7, 29, 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 (3, 14, 64)-net over F32, using
- (u, u+v)-construction [i] based on
(42−21, 42, 288)-Net in Base 32 — Constructive
(21, 42, 288)-net in base 32, using
- base change [i] based on digital (9, 30, 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
(42−21, 42, 515)-Net over F32 — Digital
Digital (21, 42, 515)-net over F32, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3242, 515, F32, 2, 21) (dual of [(515, 2), 988, 22]-NRT-code), using
- OOA 2-folding [i] based on linear OA(3242, 1030, F32, 21) (dual of [1030, 988, 22]-code), using
- construction X applied to C([0,10]) ⊂ C([0,9]) [i] based on
- 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(3237, 1025, F32, 19) (dual of [1025, 988, 20]-code), using the expurgated narrow-sense BCH-code C(I) with length 1025 | 324−1, defining interval I = [0,9], and minimum distance d ≥ |{−9,−8,…,9}|+1 = 20 (BCH-bound) [i]
- linear OA(321, 5, F32, 1) (dual of [5, 4, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(321, s, F32, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to C([0,10]) ⊂ C([0,9]) [i] based on
- OOA 2-folding [i] based on linear OA(3242, 1030, F32, 21) (dual of [1030, 988, 22]-code), using
(42−21, 42, 216630)-Net in Base 32 — Upper bound on s
There is no (21, 42, 216631)-net in base 32, because
- 1 times m-reduction [i] would yield (21, 41, 216631)-net in base 32, but
- the generalized Rao bound for nets shows that 32m ≥ 51 422907 708160 202209 688259 533265 383243 939887 578060 961036 312338 > 3241 [i]