Best Known (39, 39+30, s)-Nets in Base 32
(39, 39+30, 240)-Net over F32 — Constructive and digital
Digital (39, 69, 240)-net over F32, using
- 4 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, 39+30, 386)-Net in Base 32 — Constructive
(39, 69, 386)-net in base 32, using
- (u, u+v)-construction [i] based on
- (6, 21, 129)-net in base 32, using
- base change [i] based on digital (0, 15, 129)-net over F128, using
- net from sequence [i] based on digital (0, 128)-sequence over F128, using
- generalized Faure sequence [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F128 with g(F) = 0 and N(F) ≥ 129, using
- the rational function field F128(x) [i]
- Niederreiter sequence [i]
- net from sequence [i] based on digital (0, 128)-sequence over F128, using
- base change [i] based on digital (0, 15, 129)-net over F128, using
- (18, 48, 257)-net in base 32, using
- base change [i] based on digital (0, 30, 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, 30, 257)-net over F256, using
- (6, 21, 129)-net in base 32, using
(39, 39+30, 1451)-Net over F32 — Digital
Digital (39, 69, 1451)-net over F32, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3269, 1451, F32, 30) (dual of [1451, 1382, 31]-code), using
- 414 step Varšamov–Edel lengthening with (ri) = (4, 0, 0, 1, 5 times 0, 1, 15 times 0, 1, 37 times 0, 1, 73 times 0, 1, 120 times 0, 1, 155 times 0) [i] based on linear OA(3259, 1027, F32, 30) (dual of [1027, 968, 31]-code), using
- construction XX applied to C1 = C([1022,27]), C2 = C([0,28]), C3 = C1 + C2 = C([0,27]), and C∩ = C1 ∩ C2 = C([1022,28]) [i] based on
- linear OA(3257, 1023, F32, 29) (dual of [1023, 966, 30]-code), using the primitive BCH-code C(I) with length 1023 = 322−1, defining interval I = {−1,0,…,27}, and designed minimum distance d ≥ |I|+1 = 30 [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(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(3255, 1023, F32, 28) (dual of [1023, 968, 29]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 1023 = 322−1, defining interval I = [0,27], and designed minimum distance d ≥ |I|+1 = 29 [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([1022,27]), C2 = C([0,28]), C3 = C1 + C2 = C([0,27]), and C∩ = C1 ∩ C2 = C([1022,28]) [i] based on
- 414 step Varšamov–Edel lengthening with (ri) = (4, 0, 0, 1, 5 times 0, 1, 15 times 0, 1, 37 times 0, 1, 73 times 0, 1, 120 times 0, 1, 155 times 0) [i] based on linear OA(3259, 1027, F32, 30) (dual of [1027, 968, 31]-code), using
(39, 39+30, 1738172)-Net in Base 32 — Upper bound on s
There is no (39, 69, 1738173)-net in base 32, because
- 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]