Best Known (43−19, 43, s)-Nets in Base 32
(43−19, 43, 196)-Net over F32 — Constructive and digital
Digital (24, 43, 196)-net over F32, using
- 1 times m-reduction [i] based on digital (24, 44, 196)-net over F32, using
- (u, u+v)-construction [i] based on
- digital (7, 17, 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 (7, 27, 98)-net over F32, using
- net from sequence [i] based on digital (7, 97)-sequence over F32 (see above)
- digital (7, 17, 98)-net over F32, using
- (u, u+v)-construction [i] based on
(43−19, 43, 337)-Net in Base 32 — Constructive
(24, 43, 337)-net in base 32, using
- (u, u+v)-construction [i] based on
- (3, 12, 80)-net in base 32, using
- base change [i] based on digital (1, 10, 80)-net over F64, using
- net from sequence [i] based on digital (1, 79)-sequence over F64, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F64 with g(F) = 1 and N(F) ≥ 80, using
- net from sequence [i] based on digital (1, 79)-sequence over F64, using
- base change [i] based on digital (1, 10, 80)-net over F64, using
- (12, 31, 257)-net in base 32, using
- 1 times m-reduction [i] based on (12, 32, 257)-net in base 32, using
- base change [i] based on digital (0, 20, 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, 20, 257)-net over F256, using
- 1 times m-reduction [i] based on (12, 32, 257)-net in base 32, using
- (3, 12, 80)-net in base 32, using
(43−19, 43, 1099)-Net over F32 — Digital
Digital (24, 43, 1099)-net over F32, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3243, 1099, F32, 19) (dual of [1099, 1056, 20]-code), using
- 64 step Varšamov–Edel lengthening with (ri) = (3, 0, 0, 0, 1, 14 times 0, 1, 44 times 0) [i] based on linear OA(3238, 1030, F32, 19) (dual of [1030, 992, 20]-code), using
- construction X applied to C([0,9]) ⊂ C([0,8]) [i] based on
- 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(3233, 1025, F32, 17) (dual of [1025, 992, 18]-code), using the expurgated narrow-sense BCH-code C(I) with length 1025 | 324−1, defining interval I = [0,8], and minimum distance d ≥ |{−8,−7,…,8}|+1 = 18 (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([0,9]) ⊂ C([0,8]) [i] based on
- 64 step Varšamov–Edel lengthening with (ri) = (3, 0, 0, 0, 1, 14 times 0, 1, 44 times 0) [i] based on linear OA(3238, 1030, F32, 19) (dual of [1030, 992, 20]-code), using
(43−19, 43, 1413909)-Net in Base 32 — Upper bound on s
There is no (24, 43, 1413910)-net in base 32, because
- 1 times m-reduction [i] would yield (24, 42, 1413910)-net in base 32, but
- the generalized Rao bound for nets shows that 32m ≥ 1645 507880 684641 113083 198323 031198 566108 662421 224252 417830 320118 > 3242 [i]