Best Known (56−25, 56, s)-Nets in Base 32
(56−25, 56, 218)-Net over F32 — Constructive and digital
Digital (31, 56, 218)-net over F32, using
- 1 times m-reduction [i] based on digital (31, 57, 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, 37, 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
- (u, u+v)-construction [i] based on
(56−25, 56, 337)-Net in Base 32 — Constructive
(31, 56, 337)-net in base 32, using
- (u, u+v)-construction [i] based on
- (4, 16, 80)-net in base 32, using
- 2 times m-reduction [i] based on (4, 18, 80)-net in base 32, using
- base change [i] based on digital (1, 15, 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, 15, 80)-net over F64, using
- 2 times m-reduction [i] based on (4, 18, 80)-net in base 32, using
- (15, 40, 257)-net in base 32, using
- base change [i] based on digital (0, 25, 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, 25, 257)-net over F256, using
- (4, 16, 80)-net in base 32, using
(56−25, 56, 1123)-Net over F32 — Digital
Digital (31, 56, 1123)-net over F32, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3256, 1123, F32, 25) (dual of [1123, 1067, 26]-code), using
- 89 step Varšamov–Edel lengthening with (ri) = (4, 0, 0, 1, 8 times 0, 1, 23 times 0, 1, 52 times 0) [i] based on linear OA(3249, 1027, F32, 25) (dual of [1027, 978, 26]-code), using
- construction XX applied to C1 = C([1022,22]), C2 = C([0,23]), C3 = C1 + C2 = C([0,22]), and C∩ = C1 ∩ C2 = C([1022,23]) [i] based on
- linear OA(3247, 1023, F32, 24) (dual of [1023, 976, 25]-code), using the primitive BCH-code C(I) with length 1023 = 322−1, defining interval I = {−1,0,…,22}, and designed minimum distance d ≥ |I|+1 = 25 [i]
- linear OA(3247, 1023, F32, 24) (dual of [1023, 976, 25]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 1023 = 322−1, defining interval I = [0,23], and designed minimum distance d ≥ |I|+1 = 25 [i]
- linear OA(3249, 1023, F32, 25) (dual of [1023, 974, 26]-code), using the primitive BCH-code C(I) with length 1023 = 322−1, defining interval I = {−1,0,…,23}, and designed minimum distance d ≥ |I|+1 = 26 [i]
- linear OA(3245, 1023, F32, 23) (dual of [1023, 978, 24]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 1023 = 322−1, defining interval I = [0,22], and designed minimum distance d ≥ |I|+1 = 24 [i]
- linear OA(320, 2, F32, 0) (dual of [2, 2, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(320, s, F32, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- linear OA(320, 2, F32, 0) (dual of [2, 2, 1]-code) (see above)
- construction XX applied to C1 = C([1022,22]), C2 = C([0,23]), C3 = C1 + C2 = C([0,22]), and C∩ = C1 ∩ C2 = C([1022,23]) [i] based on
- 89 step Varšamov–Edel lengthening with (ri) = (4, 0, 0, 1, 8 times 0, 1, 23 times 0, 1, 52 times 0) [i] based on linear OA(3249, 1027, F32, 25) (dual of [1027, 978, 26]-code), using
(56−25, 56, 1350833)-Net in Base 32 — Upper bound on s
There is no (31, 56, 1350834)-net in base 32, because
- 1 times m-reduction [i] would yield (31, 55, 1350834)-net in base 32, but
- the generalized Rao bound for nets shows that 32m ≥ 60708 551666 588403 671288 565493 218602 350639 779655 938470 745491 098563 678977 393027 925972 > 3255 [i]