Best Known (39, 70, s)-Nets in Base 32
(39, 70, 240)-Net over F32 — Constructive and digital
Digital (39, 70, 240)-net over F32, using
- 3 times m-reduction [i] based on digital (39, 73, 240)-net over F32, using
- (u, u+v)-construction [i] based on
- digital (11, 28, 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 (11, 45, 120)-net over F32, using
- net from sequence [i] based on digital (11, 119)-sequence over F32 (see above)
- digital (11, 28, 120)-net over F32, using
- (u, u+v)-construction [i] based on
(39, 70, 354)-Net in Base 32 — Constructive
(39, 70, 354)-net in base 32, using
- (u, u+v)-construction [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
- (15, 46, 177)-net in base 32, using
- 2 times m-reduction [i] based on (15, 48, 177)-net in base 32, using
- base change [i] based on digital (7, 40, 177)-net over F64, using
- net from sequence [i] based on digital (7, 176)-sequence over F64, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F64 with g(F) = 7 and N(F) ≥ 177, using
- net from sequence [i] based on digital (7, 176)-sequence over F64, using
- base change [i] based on digital (7, 40, 177)-net over F64, using
- 2 times m-reduction [i] based on (15, 48, 177)-net in base 32, using
- (9, 24, 257)-net in base 32, using
(39, 70, 1283)-Net over F32 — Digital
Digital (39, 70, 1283)-net over F32, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3270, 1283, F32, 31) (dual of [1283, 1213, 32]-code), using
- 247 step Varšamov–Edel lengthening with (ri) = (4, 0, 1, 6 times 0, 1, 15 times 0, 1, 34 times 0, 1, 71 times 0, 1, 114 times 0) [i] based on linear OA(3261, 1027, F32, 31) (dual of [1027, 966, 32]-code), using
- construction XX applied to C1 = C([1022,28]), C2 = C([0,29]), C3 = C1 + C2 = C([0,28]), and C∩ = C1 ∩ C2 = C([1022,29]) [i] based on
- linear OA(3259, 1023, F32, 30) (dual of [1023, 964, 31]-code), using the primitive BCH-code C(I) with length 1023 = 322−1, defining interval I = {−1,0,…,28}, and designed minimum distance d ≥ |I|+1 = 31 [i]
- linear OA(3259, 1023, F32, 30) (dual of [1023, 964, 31]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 1023 = 322−1, defining interval I = [0,29], and designed minimum distance d ≥ |I|+1 = 31 [i]
- linear OA(3261, 1023, F32, 31) (dual of [1023, 962, 32]-code), using the primitive BCH-code C(I) with length 1023 = 322−1, defining interval I = {−1,0,…,29}, and designed minimum distance d ≥ |I|+1 = 32 [i]
- linear OA(3257, 1023, F32, 29) (dual of [1023, 966, 30]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 1023 = 322−1, defining interval I = [0,28], and designed minimum distance d ≥ |I|+1 = 30 [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([1022,28]), C2 = C([0,29]), C3 = C1 + C2 = C([0,28]), and C∩ = C1 ∩ C2 = C([1022,29]) [i] based on
- 247 step Varšamov–Edel lengthening with (ri) = (4, 0, 1, 6 times 0, 1, 15 times 0, 1, 34 times 0, 1, 71 times 0, 1, 114 times 0) [i] based on linear OA(3261, 1027, F32, 31) (dual of [1027, 966, 32]-code), using
(39, 70, 1738172)-Net in Base 32 — Upper bound on s
There is no (39, 70, 1738173)-net in base 32, because
- 1 times m-reduction [i] would yield (39, 69, 1738173)-net in base 32, but
- the generalized Rao bound for nets shows that 32m ≥ 71 671886 744410 316923 420046 879217 390308 388339 056904 369953 865078 520729 799158 589839 283857 331750 477160 482416 > 3269 [i]