Best Known (52−23, 52, s)-Nets in Base 32
(52−23, 52, 218)-Net over F32 — Constructive and digital
Digital (29, 52, 218)-net over F32, using
- (u, u+v)-construction [i] based on
- digital (7, 18, 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, 34, 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, 18, 98)-net over F32, using
(52−23, 52, 322)-Net in Base 32 — Constructive
(29, 52, 322)-net in base 32, using
- (u, u+v)-construction [i] based on
- (3, 14, 65)-net in base 32, using
- 4 times m-reduction [i] based on (3, 18, 65)-net in base 32, using
- base change [i] based on digital (0, 15, 65)-net over F64, using
- net from sequence [i] based on digital (0, 64)-sequence over F64, using
- generalized Faure sequence [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F64 with g(F) = 0 and N(F) ≥ 65, using
- the rational function field F64(x) [i]
- Niederreiter sequence [i]
- net from sequence [i] based on digital (0, 64)-sequence over F64, using
- base change [i] based on digital (0, 15, 65)-net over F64, using
- 4 times m-reduction [i] based on (3, 18, 65)-net in base 32, using
- (15, 38, 257)-net in base 32, using
- 2 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
- 2 times m-reduction [i] based on (15, 40, 257)-net in base 32, using
- (3, 14, 65)-net in base 32, using
(52−23, 52, 1142)-Net over F32 — Digital
Digital (29, 52, 1142)-net over F32, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3252, 1142, F32, 23) (dual of [1142, 1090, 24]-code), using
- 108 step Varšamov–Edel lengthening with (ri) = (4, 0, 0, 1, 10 times 0, 1, 27 times 0, 1, 65 times 0) [i] based on linear OA(3245, 1027, F32, 23) (dual of [1027, 982, 24]-code), using
- construction XX applied to C1 = C([1022,20]), C2 = C([0,21]), C3 = C1 + C2 = C([0,20]), and C∩ = C1 ∩ C2 = C([1022,21]) [i] based on
- linear OA(3243, 1023, F32, 22) (dual of [1023, 980, 23]-code), using the primitive BCH-code C(I) with length 1023 = 322−1, defining interval I = {−1,0,…,20}, and designed minimum distance d ≥ |I|+1 = 23 [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(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(3241, 1023, F32, 21) (dual of [1023, 982, 22]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 1023 = 322−1, defining interval I = [0,20], and designed minimum distance d ≥ |I|+1 = 22 [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([1022,20]), C2 = C([0,21]), C3 = C1 + C2 = C([0,20]), and C∩ = C1 ∩ C2 = C([1022,21]) [i] based on
- 108 step Varšamov–Edel lengthening with (ri) = (4, 0, 0, 1, 10 times 0, 1, 27 times 0, 1, 65 times 0) [i] based on linear OA(3245, 1027, F32, 23) (dual of [1027, 982, 24]-code), using
(52−23, 52, 1506862)-Net in Base 32 — Upper bound on s
There is no (29, 52, 1506863)-net in base 32, because
- 1 times m-reduction [i] would yield (29, 51, 1506863)-net in base 32, but
- the generalized Rao bound for nets shows that 32m ≥ 57896 372240 526740 965902 919432 904747 580898 943805 390330 451206 417653 156898 838532 > 3251 [i]