Best Known (19, 34, s)-Nets in Base 32
(19, 34, 196)-Net over F32 — Constructive and digital
Digital (19, 34, 196)-net over F32, using
- (u, u+v)-construction [i] based on
- digital (5, 12, 98)-net over F32, using
- s-reduction based on digital (5, 12, 99)-net over F32, using
- generalized (u, u+v)-construction [i] based on
- digital (0, 2, 33)-net over F32, using
- 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, 7, 33)-net over F32, using
- net from sequence [i] based on digital (0, 32)-sequence over F32 (see above)
- generalized (u, u+v)-construction [i] based on
- s-reduction based on digital (5, 12, 99)-net over F32, using
- digital (7, 22, 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 (5, 12, 98)-net over F32, using
(19, 34, 337)-Net in Base 32 — Constructive
(19, 34, 337)-net in base 32, using
- (u, u+v)-construction [i] based on
- (3, 10, 80)-net in base 32, using
- 2 times m-reduction [i] based on (3, 12, 80)-net in base 32, using
- base change [i] based on digital (1, 10, 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, 10, 80)-net over F64, using
- 2 times m-reduction [i] based on (3, 12, 80)-net in base 32, using
- (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
- (3, 10, 80)-net in base 32, using
(19, 34, 1095)-Net over F32 — Digital
Digital (19, 34, 1095)-net over F32, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3234, 1095, F32, 15) (dual of [1095, 1061, 16]-code), using
- 63 step Varšamov–Edel lengthening with (ri) = (3, 0, 0, 0, 1, 13 times 0, 1, 44 times 0) [i] based on linear OA(3229, 1027, F32, 15) (dual of [1027, 998, 16]-code), using
- construction XX applied to C1 = C([1022,12]), C2 = C([0,13]), C3 = C1 + C2 = C([0,12]), and C∩ = C1 ∩ C2 = C([1022,13]) [i] based on
- 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(3227, 1023, F32, 14) (dual of [1023, 996, 15]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 1023 = 322−1, defining interval I = [0,13], and designed minimum distance d ≥ |I|+1 = 15 [i]
- linear OA(3229, 1023, F32, 15) (dual of [1023, 994, 16]-code), using the primitive BCH-code C(I) with length 1023 = 322−1, defining interval I = {−1,0,…,13}, and designed minimum distance d ≥ |I|+1 = 16 [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(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([1022,12]), C2 = C([0,13]), C3 = C1 + C2 = C([0,12]), and C∩ = C1 ∩ C2 = C([1022,13]) [i] based on
- 63 step Varšamov–Edel lengthening with (ri) = (3, 0, 0, 0, 1, 13 times 0, 1, 44 times 0) [i] based on linear OA(3229, 1027, F32, 15) (dual of [1027, 998, 16]-code), using
(19, 34, 1359136)-Net in Base 32 — Upper bound on s
There is no (19, 34, 1359137)-net in base 32, because
- 1 times m-reduction [i] would yield (19, 33, 1359137)-net in base 32, but
- the generalized Rao bound for nets shows that 32m ≥ 46 768256 966218 105509 235865 831123 082444 775352 861472 > 3233 [i]