Best Known (18, 18+14, s)-Nets in Base 32
(18, 18+14, 198)-Net over F32 — Constructive and digital
Digital (18, 32, 198)-net over F32, using
- generalized (u, u+v)-construction [i] based on
- digital (0, 2, 33)-net over F32, using
- digital (0, 2, 33)-net over F32 (see above)
- 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, 4, 33)-net over F32, using
- net from sequence [i] based on digital (0, 32)-sequence over F32 (see above)
- digital (0, 7, 33)-net over F32, using
- net from sequence [i] based on digital (0, 32)-sequence over F32 (see above)
- digital (0, 14, 33)-net over F32, using
- net from sequence [i] based on digital (0, 32)-sequence over F32 (see above)
(18, 18+14, 322)-Net in Base 32 — Constructive
(18, 32, 322)-net in base 32, using
- (u, u+v)-construction [i] based on
- (2, 9, 65)-net in base 32, using
- 3 times m-reduction [i] based on (2, 12, 65)-net in base 32, using
- base change [i] based on digital (0, 10, 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, 10, 65)-net over F64, using
- 3 times m-reduction [i] based on (2, 12, 65)-net in base 32, using
- (9, 23, 257)-net in base 32, using
- 1 times m-reduction [i] based on (9, 24, 257)-net in base 32, using
- base change [i] based on digital (0, 15, 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, 15, 257)-net over F256, using
- 1 times m-reduction [i] based on (9, 24, 257)-net in base 32, using
- (2, 9, 65)-net in base 32, using
(18, 18+14, 1119)-Net over F32 — Digital
Digital (18, 32, 1119)-net over F32, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3232, 1119, F32, 14) (dual of [1119, 1087, 15]-code), using
- 87 step Varšamov–Edel lengthening with (ri) = (3, 4 times 0, 1, 19 times 0, 1, 61 times 0) [i] based on linear OA(3227, 1027, F32, 14) (dual of [1027, 1000, 15]-code), using
- construction XX applied to C1 = C([1022,11]), C2 = C([0,12]), C3 = C1 + C2 = C([0,11]), and C∩ = C1 ∩ C2 = C([1022,12]) [i] based on
- linear OA(3225, 1023, F32, 13) (dual of [1023, 998, 14]-code), using the primitive BCH-code C(I) with length 1023 = 322−1, defining interval I = {−1,0,…,11}, and designed minimum distance d ≥ |I|+1 = 14 [i]
- linear OA(3225, 1023, F32, 13) (dual of [1023, 998, 14]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 1023 = 322−1, defining interval I = [0,12], and designed minimum distance d ≥ |I|+1 = 14 [i]
- linear OA(3227, 1023, F32, 14) (dual of [1023, 996, 15]-code), using the primitive BCH-code C(I) with length 1023 = 322−1, defining interval I = {−1,0,…,12}, and designed minimum distance d ≥ |I|+1 = 15 [i]
- linear OA(3223, 1023, F32, 12) (dual of [1023, 1000, 13]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 1023 = 322−1, defining interval I = [0,11], and designed minimum distance d ≥ |I|+1 = 13 [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([1022,11]), C2 = C([0,12]), C3 = C1 + C2 = C([0,11]), and C∩ = C1 ∩ C2 = C([1022,12]) [i] based on
- 87 step Varšamov–Edel lengthening with (ri) = (3, 4 times 0, 1, 19 times 0, 1, 61 times 0) [i] based on linear OA(3227, 1027, F32, 14) (dual of [1027, 1000, 15]-code), using
(18, 18+14, 828401)-Net in Base 32 — Upper bound on s
There is no (18, 32, 828402)-net in base 32, because
- the generalized Rao bound for nets shows that 32m ≥ 1 461507 773148 847639 145862 123705 489009 860969 372216 > 3232 [i]