Best Known (57−24, 57, s)-Nets in Base 32
(57−24, 57, 228)-Net over F32 — Constructive and digital
Digital (33, 57, 228)-net over F32, using
- (u, u+v)-construction [i] based on
- digital (10, 22, 108)-net over F32, using
- (u, u+v)-construction [i] based on
- digital (1, 7, 44)-net over F32, using
- net from sequence [i] based on digital (1, 43)-sequence over F32, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F32 with g(F) = 1 and N(F) ≥ 44, using
- net from sequence [i] based on digital (1, 43)-sequence over F32, using
- digital (3, 15, 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 (1, 7, 44)-net over F32, using
- (u, u+v)-construction [i] based on
- digital (11, 35, 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 (10, 22, 108)-net over F32, using
(57−24, 57, 386)-Net in Base 32 — Constructive
(33, 57, 386)-net in base 32, using
- (u, u+v)-construction [i] based on
- (6, 18, 129)-net in base 32, using
- 3 times m-reduction [i] based on (6, 21, 129)-net in base 32, using
- base change [i] based on digital (0, 15, 129)-net over F128, using
- net from sequence [i] based on digital (0, 128)-sequence over F128, using
- generalized Faure sequence [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F128 with g(F) = 0 and N(F) ≥ 129, using
- the rational function field F128(x) [i]
- Niederreiter sequence [i]
- net from sequence [i] based on digital (0, 128)-sequence over F128, using
- base change [i] based on digital (0, 15, 129)-net over F128, using
- 3 times m-reduction [i] based on (6, 21, 129)-net in base 32, using
- (15, 39, 257)-net in base 32, using
- 1 times m-reduction [i] based on (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
- 1 times m-reduction [i] based on (15, 40, 257)-net in base 32, using
- (6, 18, 129)-net in base 32, using
(57−24, 57, 1646)-Net over F32 — Digital
Digital (33, 57, 1646)-net over F32, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3257, 1646, F32, 24) (dual of [1646, 1589, 25]-code), using
- 609 step Varšamov–Edel lengthening with (ri) = (4, 0, 0, 1, 9 times 0, 1, 25 times 0, 1, 57 times 0, 1, 113 times 0, 1, 174 times 0, 1, 222 times 0) [i] based on linear OA(3247, 1027, F32, 24) (dual of [1027, 980, 25]-code), using
- construction XX applied to C1 = C([1022,21]), C2 = C([0,22]), C3 = C1 + C2 = C([0,21]), and C∩ = C1 ∩ C2 = C([1022,22]) [i] based on
- linear OA(3245, 1023, F32, 23) (dual of [1023, 978, 24]-code), using the primitive BCH-code C(I) with length 1023 = 322−1, defining interval I = {−1,0,…,21}, and designed minimum distance d ≥ |I|+1 = 24 [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(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(3243, 1023, F32, 22) (dual of [1023, 980, 23]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 1023 = 322−1, defining interval I = [0,21], and designed minimum distance d ≥ |I|+1 = 23 [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([1022,21]), C2 = C([0,22]), C3 = C1 + C2 = C([0,21]), and C∩ = C1 ∩ C2 = C([1022,22]) [i] based on
- 609 step Varšamov–Edel lengthening with (ri) = (4, 0, 0, 1, 9 times 0, 1, 25 times 0, 1, 57 times 0, 1, 113 times 0, 1, 174 times 0, 1, 222 times 0) [i] based on linear OA(3247, 1027, F32, 24) (dual of [1027, 980, 25]-code), using
(57−24, 57, 2406916)-Net in Base 32 — Upper bound on s
There is no (33, 57, 2406917)-net in base 32, because
- the generalized Rao bound for nets shows that 32m ≥ 62 165507 146564 793302 283904 899669 494761 337535 549070 292604 881645 999680 019366 730161 767232 > 3257 [i]