Best Known (30, 30+23, s)-Nets in Base 32
(30, 30+23, 219)-Net over F32 — Constructive and digital
Digital (30, 53, 219)-net over F32, using
- (u, u+v)-construction [i] based on
- digital (8, 19, 99)-net over F32, using
- generalized (u, u+v)-construction [i] based on
- digital (0, 3, 33)-net over F32, using
- net from sequence [i] based on digital (0, 32)-sequence over F32, using
- generalized Faure sequence [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F32 with g(F) = 0 and N(F) ≥ 33, using
- the rational function field F32(x) [i]
- Niederreiter sequence [i]
- net from sequence [i] based on digital (0, 32)-sequence over F32, using
- digital (0, 5, 33)-net over F32, using
- net from sequence [i] based on digital (0, 32)-sequence over F32 (see above)
- digital (0, 11, 33)-net over F32, using
- net from sequence [i] based on digital (0, 32)-sequence over F32 (see above)
- digital (0, 3, 33)-net over F32, using
- generalized (u, u+v)-construction [i] based on
- 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 (8, 19, 99)-net over F32, using
(30, 30+23, 337)-Net in Base 32 — Constructive
(30, 53, 337)-net in base 32, using
- (u, u+v)-construction [i] based on
- (4, 15, 80)-net in base 32, using
- 3 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
- 3 times m-reduction [i] based on (4, 18, 80)-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
- (4, 15, 80)-net in base 32, using
(30, 30+23, 1269)-Net over F32 — Digital
Digital (30, 53, 1269)-net over F32, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3253, 1269, F32, 23) (dual of [1269, 1216, 24]-code), using
- 234 step Varšamov–Edel lengthening with (ri) = (4, 0, 0, 1, 10 times 0, 1, 27 times 0, 1, 65 times 0, 1, 125 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
- 234 step Varšamov–Edel lengthening with (ri) = (4, 0, 0, 1, 10 times 0, 1, 27 times 0, 1, 65 times 0, 1, 125 times 0) [i] based on linear OA(3245, 1027, F32, 23) (dual of [1027, 982, 24]-code), using
(30, 30+23, 2064932)-Net in Base 32 — Upper bound on s
There is no (30, 53, 2064933)-net in base 32, because
- 1 times m-reduction [i] would yield (30, 52, 2064933)-net in base 32, but
- the generalized Rao bound for nets shows that 32m ≥ 1 852682 144545 908954 643203 933105 945636 754472 742389 796547 076064 827857 219545 462352 > 3252 [i]