Cyclic Polar Codes Narayanan Rengaswamy Henry D. Pfister Texas - - PowerPoint PPT Presentation
Cyclic Polar Codes Narayanan Rengaswamy Henry D. Pfister Texas A&M University Duke University r narayanan 92@tamu.edu henry.pfister@duke.edu 2015 IEEE International Symposium on Information Theory June 16, 2015 Narayanan Rengaswamy
Narayanan Rengaswamy
Texas A&M University
r narayanan 92@tamu.edu Henry D. Pfister
Duke University
henry.pfister@duke.edu
2015 IEEE International Symposium on Information Theory June 16, 2015
Narayanan Rengaswamy & Henry D. Pfister Cyclic Polar Codes June 16, 2015 1 / 32
Polar Codes
1
Polar Codes Overview
2
Cyclic Polar Codes Galois field Fourier transform Cyclic polar codes
3
Decoding Successive Cancellation Decoding
4
Results For the q-ary Erasure Channel For the q-ary Symmetric Channel
5
Conclusions
Narayanan Rengaswamy & Henry D. Pfister Cyclic Polar Codes June 16, 2015 2 / 32
Polar Codes Overview
Introduced by Arıkan in [Arı09] using the binary 2 × 2 kernel G2 = 1 1 1
GN = BNG ⊗n
2 ,
where BN is the length-N bit-reversal permutation matrix. Shown to achieve the symmetric capacity of binary input DMCs, asymptotically, under successive cancellation (SC) decoding.
Narayanan Rengaswamy & Henry D. Pfister Cyclic Polar Codes June 16, 2015 3 / 32
Polar Codes Overview
Blocklength N = ℓn, ℓ > 2; Transformation GN = BNG ⊗n
ℓ
Korada et al.: Binary Gℓ for binary DMCs [KS ¸U10]. S ¸a¸ so˘ glu et al.: Binary Gℓ for q-ary DMCs, q prime [S ¸TA09].
Narayanan Rengaswamy & Henry D. Pfister Cyclic Polar Codes June 16, 2015 4 / 32
Polar Codes Overview
Blocklength N = ℓn, ℓ > 2; Transformation GN = BNG ⊗n
ℓ
Korada et al.: Binary Gℓ for binary DMCs [KS ¸U10]. S ¸a¸ so˘ glu et al.: Binary Gℓ for q-ary DMCs, q prime [S ¸TA09]. Mori and Tanaka: Non-binary Gℓ for arbitrary q-ary DMCs, q = pm [MT10; MT14].
For example, using extended RS matrices. GRS(3, 3) = 1 1 2 1 1 1 1 = α0 α0 α1 α0 α0 α0 α0 where α = 2 ∈ F3 is primitive.
Narayanan Rengaswamy & Henry D. Pfister Cyclic Polar Codes June 16, 2015 4 / 32
Polar Codes Overview
So far, blocklength N = ℓn and transformation GN = BNG ⊗n
ℓ
.
Narayanan Rengaswamy & Henry D. Pfister Cyclic Polar Codes June 16, 2015 5 / 32
Polar Codes Overview
System implementation: Many systems use RS or other cyclic codes.
Narayanan Rengaswamy & Henry D. Pfister Cyclic Polar Codes June 16, 2015 6 / 32
CP Codes
1
Polar Codes Overview
2
Cyclic Polar Codes Galois field Fourier transform Cyclic polar codes
3
Decoding Successive Cancellation Decoding
4
Results For the q-ary Erasure Channel For the q-ary Symmetric Channel
5
Conclusions
Narayanan Rengaswamy & Henry D. Pfister Cyclic Polar Codes June 16, 2015 7 / 32
CP Codes Galois field Fourier transform
Replace polar transform with Galois field Fourier Transform (GFFT). Input: u = (u0, u1, . . . , uN−1), Output: v = (v0, v1, . . . , vN−1). Then u
GFFT
← − − − v If FN is the GFFT matrix, u = FNv (or) uj =
N−1
viωij for j = 0, 1, . . . , N − 1, where ω in Fq has order N.
Narayanan Rengaswamy & Henry D. Pfister Cyclic Polar Codes June 16, 2015 8 / 32
CP Codes Galois field Fourier transform
How is this related to the polar transform?
Narayanan Rengaswamy & Henry D. Pfister Cyclic Polar Codes June 16, 2015 9 / 32
CP Codes Galois field Fourier transform
How is this related to the polar transform? Evaluate the GFFT using the Cooley-Tukey FFT [CT65].
Factor N = n
m=1 ℓm = ℓ1ℓ2 · · · ℓn.
Implement small GFFTs of length ℓm directly. Combine them using appropriate twiddle factors and index-shuffles. Simplest case is ℓ1 = ℓ2 = · · · = ℓn = 2 for N = 2n; equivalent to standard polar code.
Ignoring twiddle factors, the Kronecker product of repeated short GFFTs gives a long GFFT.
Narayanan Rengaswamy & Henry D. Pfister Cyclic Polar Codes June 16, 2015 9 / 32
CP Codes Galois field Fourier transform
− → Encode u(0)
i
Stage m = 0 ℓ1 = 5 F ′
ℓ1
Stage m = 1 u(1)
i
ℓ2 = 3 F ′
ℓ2
Stage m = 2 ← − Decode u0 u1 u2 u3 u4 u5 u6 u7 u8 u9 u10 u11 u12 u13 u14 u0 u3 u6 u9 u12 u1 u4 u7 u10 u13 u2 u5 u8 u11 u14 F ′
5
F ′
5
F ′
5
1 α−1 α−2 α−3 α−4 1 α−2 α−4 α−6 α−8 F ′
3
F ′
3
F ′
3
F ′
3
F ′
3
v0 v5 v10 v1 v6 v11 v2 v7 v12 v3 v8 v13 v4 v9 v14 v0 v1 v2 v3 v4 v5 v6 v7 v8 v9 v10 v11 v12 v13 v14 GFFT
Input Shuffle Length-5 IFFTs Index Shuffle Length-3 IFFTs Output Shuffle GIFFT
An example for N = 15 over F16 depicting the transform. α is a primitive element in F16. In this case ω = α.
Narayanan Rengaswamy & Henry D. Pfister Cyclic Polar Codes June 16, 2015 10 / 32
CP Codes Cyclic polar codes
Recollect that uj =
N−1
viωij where ωN = 1 in Fq. In polynomial notation, with v(x) = N−1
i=0 vixi, we have
u(x) =
N−1
ujxj =
N−1
v(ωj)xj Hence, uj’s are evaluations of v(x).
Narayanan Rengaswamy & Henry D. Pfister Cyclic Polar Codes June 16, 2015 11 / 32
CP Codes Cyclic polar codes
Design of the code C produces the set of information indices A.
Narayanan Rengaswamy & Henry D. Pfister Cyclic Polar Codes June 16, 2015 12 / 32
CP Codes Cyclic polar codes
Design of the code C produces the set of information indices A. Given Ac, the indices frozen to zeros in u(x), there exists a generator g(x) such that v(x) = uA(x)g(x) = uA(x)
(x − ωj) where ωN = 1 in Fq.
Narayanan Rengaswamy & Henry D. Pfister Cyclic Polar Codes June 16, 2015 12 / 32
CP Codes Cyclic polar codes
Design of the code C produces the set of information indices A. Given Ac, the indices frozen to zeros in u(x), there exists a generator g(x) such that v(x) = uA(x)g(x) = uA(x)
(x − ωj) where ωN = 1 in Fq. We have a cyclic code!
Narayanan Rengaswamy & Henry D. Pfister Cyclic Polar Codes June 16, 2015 12 / 32
CP Codes Cyclic polar codes
Design of the code C produces the set of information indices A. Given Ac, the indices frozen to zeros in u(x), there exists a generator g(x) such that v(x) = uA(x)g(x) = uA(x)
(x − ωj) where ωN = 1 in Fq. We have a cyclic code! Constraint: N|(q − 1). Hence, field size must grow with the blocklength.
Narayanan Rengaswamy & Henry D. Pfister Cyclic Polar Codes June 16, 2015 12 / 32
CP Codes Cyclic polar codes
Narayanan Rengaswamy & Henry D. Pfister Cyclic Polar Codes June 16, 2015 13 / 32
CP Codes Cyclic polar codes
Example: N = 3 and N = 5 over F16. G3 = 1 1 1 1 ω ω2 1 ω2 ω and G5 = 1 1 1 1 1 1 ω ω2 ω3 ω4 1 ω2 ω4 ω ω3 1 ω3 ω ω4 ω2 1 ω4 ω3 ω2 ω where ω3 = 1 for G3, ω5 = 1 for G5.
Narayanan Rengaswamy & Henry D. Pfister Cyclic Polar Codes June 16, 2015 14 / 32
CP Codes Cyclic polar codes
Example: N = 3 and N = 5 over F16. G3 = 1 1 1 1 ω ω2 1 ω2 ω and G5 = 1 1 1 1 1 1 ω ω2 ω3 ω4 1 ω2 ω4 ω ω3 1 ω3 ω ω4 ω2 1 ω4 ω3 ω2 ω where ω3 = 1 for G3, ω5 = 1 for G5. The transformation GN, a GFFT matrix, polarizes any q-ary channel because it
is invertible and not upper triangular. contains a primitive element [MT14].
Narayanan Rengaswamy & Henry D. Pfister Cyclic Polar Codes June 16, 2015 14 / 32
Decoding
1
Polar Codes Overview
2
Cyclic Polar Codes Galois field Fourier transform Cyclic polar codes
3
Decoding Successive Cancellation Decoding
4
Results For the q-ary Erasure Channel For the q-ary Symmetric Channel
5
Conclusions
Narayanan Rengaswamy & Henry D. Pfister Cyclic Polar Codes June 16, 2015 15 / 32
Decoding Successive Cancellation Decoding
Blocklength N = 2n over q-ary channel, q prime.
Narayanan Rengaswamy & Henry D. Pfister Cyclic Polar Codes June 16, 2015 16 / 32
Decoding Successive Cancellation Decoding
Blocklength N = 2n over q-ary channel, q prime. Consider a 2 × 2 butterfly: inputs (a0, a1) and outputs (b0, b1). b0 = a0 + a1 b1 = a0 + αa1
Narayanan Rengaswamy & Henry D. Pfister Cyclic Polar Codes June 16, 2015 16 / 32
Decoding Successive Cancellation Decoding
Blocklength N = 2n over q-ary channel, q prime. Consider a 2 × 2 butterfly: inputs (a0, a1) and outputs (b0, b1). b0 = a0 + a1 b1 = a0 + αa1 To estimate (a0, a1) from (b0, b1) in the polar decoding order, we have ˆ a0 = (1 − α)−1(b1 − αb0)
Narayanan Rengaswamy & Henry D. Pfister Cyclic Polar Codes June 16, 2015 16 / 32
Decoding Successive Cancellation Decoding
Blocklength N = 2n over q-ary channel, q prime. Consider a 2 × 2 butterfly: inputs (a0, a1) and outputs (b0, b1). b0 = a0 + a1 b1 = a0 + αa1 To estimate (a0, a1) from (b0, b1) in the polar decoding order, we have ˆ a0 = (1 − α)−1(b1 − αb0) ˆ a1 = b0 − a0
Narayanan Rengaswamy & Henry D. Pfister Cyclic Polar Codes June 16, 2015 16 / 32
Decoding Successive Cancellation Decoding
Blocklength N = 2n over q-ary channel, q prime. Consider a 2 × 2 butterfly: inputs (a0, a1) and outputs (b0, b1). b0 = a0 + a1 b1 = a0 + αa1 To estimate (a0, a1) from (b0, b1) in the polar decoding order, we have ˆ a0 = (1 − α)−1(b1 − αb0) ˆ a1 = b0 − a0 ˆ a′
1 = α−1(b1 − a0)
Narayanan Rengaswamy & Henry D. Pfister Cyclic Polar Codes June 16, 2015 16 / 32
Decoding Successive Cancellation Decoding
2 × 2 butterfly: Compute optimal soft estimates for (a0, a1) from (b0, b1) using standard techniques from LDPC codes.
Narayanan Rengaswamy & Henry D. Pfister Cyclic Polar Codes June 16, 2015 17 / 32
Decoding Successive Cancellation Decoding
2 × 2 butterfly: Compute optimal soft estimates for (a0, a1) from (b0, b1) using standard techniques from LDPC codes. ℓ × ℓ butterfly, ℓ > 2: Hard to implement APP decoder for general length-ℓ code over Fq.
Narayanan Rengaswamy & Henry D. Pfister Cyclic Polar Codes June 16, 2015 17 / 32
Decoding Successive Cancellation Decoding
2 × 2 butterfly: Compute optimal soft estimates for (a0, a1) from (b0, b1) using standard techniques from LDPC codes. ℓ × ℓ butterfly, ℓ > 2: Hard to implement APP decoder for general length-ℓ code over Fq. ℓ × ℓ butterfly, ℓ > 2: Alternatively, use algebraic hard-decision decoding.
Narayanan Rengaswamy & Henry D. Pfister Cyclic Polar Codes June 16, 2015 17 / 32
Decoding Successive Cancellation Decoding
q-ary Erasure Channel (QEC) with ǫ = 0.5. Use Forney’s algebraic decoder. ui ∈ F16, vi ∈ F16 ∪ {?}
u(m−1)
i
ℓm = 3 u(m)
i
u0 u1 u2 F ′
3
v0 v1 v2
GFFT
(I)FFT GIFFT
Uncover next (unknown) input using outputs and recovered inputs.
Narayanan Rengaswamy & Henry D. Pfister Cyclic Polar Codes June 16, 2015 18 / 32
Decoding Successive Cancellation Decoding
q-ary Erasure Channel (QEC) with ǫ = 0.5. Use Forney’s algebraic decoder. ui ∈ F16, vi ∈ F16 ∪ {?}, ǫ = P{vi =?} = 0.5
u(m−1)
i
ℓm = 3 u(m)
i
u0 u1 u2 F ′
3
v0
0.5
v1
0.5
v2
0.5
GFFT
(I)FFT GIFFT
Uncover next (unknown) input using outputs and recovered inputs.
Narayanan Rengaswamy & Henry D. Pfister Cyclic Polar Codes June 16, 2015 18 / 32
Decoding Successive Cancellation Decoding
q-ary Erasure Channel (QEC) with ǫ = 0.5. Use Forney’s algebraic decoder. ui ∈ F16, vi ∈ F16 ∪ {?}, ǫ = P{vi =?} = 0.5 Theoretical 1 − (1 − ǫ)3
u(m−1)
i
ℓm = 3 u(m)
i
u0
0.875
u1 u2 F ′
3
v0
0.5
v1
0.5
v2
0.5
GFFT
(I)FFT GIFFT
Uncover next (unknown) input using outputs and recovered inputs.
Narayanan Rengaswamy & Henry D. Pfister Cyclic Polar Codes June 16, 2015 18 / 32
Decoding Successive Cancellation Decoding
q-ary Erasure Channel (QEC) with ǫ = 0.5. Use Forney’s algebraic decoder. ui ∈ F16, vi ∈ F16 ∪ {?}, ǫ = P{vi =?} = 0.5 Theoretical 1 − (1 − ǫ)3 1 − [3(1 − ǫ)2ǫ + (1 − ǫ)3] ...
u(m−1)
i
ℓm = 3 u(m)
i
u0
0.875
u1
0.5
u2 F ′
3
v0
0.5
v1
0.5
v2
0.5
GFFT
(I)FFT GIFFT
Uncover next (unknown) input using outputs and recovered inputs.
Narayanan Rengaswamy & Henry D. Pfister Cyclic Polar Codes June 16, 2015 18 / 32
Decoding Successive Cancellation Decoding
q-ary Erasure Channel (QEC) with ǫ = 0.5. Use Forney’s algebraic decoder. ui ∈ F16, vi ∈ F16 ∪ {?}, ǫ = P{vi =?} = 0.5 Theoretical 1 − (1 − ǫ)3 1 − [3(1 − ǫ)2ǫ + (1 − ǫ)3] ǫ3
u(m−1)
i
ℓm = 3 u(m)
i
u0
0.875
u1
0.5
u2
0.125
F ′
3
v0
0.5
v1
0.5
v2
0.5
GFFT
(I)FFT GIFFT
Uncover next (unknown) input using outputs and recovered inputs.
Narayanan Rengaswamy & Henry D. Pfister Cyclic Polar Codes June 16, 2015 18 / 32
Decoding Successive Cancellation Decoding
− → Encode u(0)
i
Stage m = 0 ℓ1 = 5 F ′
ℓ1
Stage m = 1 u(1)
i
ℓ2 = 3 F ′
ℓ2
Stage m = 2 ← − Decode u0 u1 u2 u3 u4 u5 u6 u7 u8
(D)
u9 u10 u11
(D)
u12 u13
(D)
u14
(D)
u0 1.000 u3 0.999 u6 0.984 u9 0.879 u12 0.513 u1 0.969 u4 0.813 u7 0.500 u10 0.188
(D)u13
0.031 u2 0.487 u5 0.121
(D)u8
0.016
(D)u11
0.001
(D)u14
0.000 F ′
5
F ′
5
F ′
5
0.875 0.5 0.125 0.875 0.5 0.125 1 0.875 α−1 0.5 α−2 0.125 α−3 0.875 α−4 0.5 1 0.125 α−2 0.875 α−4 0.5 α−6 0.125 α−8 F ′
3
F ′
3
F ′
3
F ′
3
F ′
3
v0 0.5 v5 0.5 v10 0.5 v1 0.5 v6 0.5 v11 0.5 v2 0.5 v7 0.5 v12 0.5 v3 0.5 v8 0.5 v13 0.5 v4 0.5 v9 0.5 v14 0.5 v0 v1 v2 v3 v4 v5 v6 v7 v8 v9 v10 v11 v12 v13 v14 GFFT
Input Shuffle Length-5 IFFTs Index Shuffle Length-3 IFFTs Output Shuffle GIFFT
Narayanan Rengaswamy & Henry D. Pfister Cyclic Polar Codes June 16, 2015 19 / 32
Decoding Successive Cancellation Decoding
q-ary Symmetric Channel with Erasures (QSCE) with e = 0.5, ǫ = 0. Use Berlekamp-Massey (B-M) and Forney algorithms. ui ∈ F16, vi ∈ F16 ∪ {?}
u(m−1)
i
ℓm = 3 u(m)
i
u0 u1 u2 F ′
3
v0 v1 v2
GFFT
(I)FFT GIFFT
Uncover next (unknown) input using outputs and recovered inputs.
Narayanan Rengaswamy & Henry D. Pfister Cyclic Polar Codes June 16, 2015 20 / 32
Decoding Successive Cancellation Decoding
q-ary Symmetric Channel with Erasures (QSCE) with e = 0.5, ǫ = 0. Use Berlekamp-Massey (B-M) and Forney algorithms. ui ∈ F16, vi ∈ F16 ∪ {?}, e = P{vi = u(m)
i
} = 0.5, ǫ = P{vi =?} = 0
u(m−1)
i
ℓm = 3 u(m)
i
u0 u1 u2 F ′
3
v0
0.5
v1
0.5
v2
0.5
GFFT
(I)FFT GIFFT
Uncover next (unknown) input using outputs and recovered inputs.
Narayanan Rengaswamy & Henry D. Pfister Cyclic Polar Codes June 16, 2015 20 / 32
Decoding Successive Cancellation Decoding
q-ary Symmetric Channel with Erasures (QSCE) with e = 0.5, ǫ = 0. Use Berlekamp-Massey (B-M) and Forney algorithms. ui ∈ F16, vi ∈ F16 ∪ {?}, e = P{vi = u(m)
i
} = 0.5, ǫ = P{vi =?} = 0 eu0 = 0.846, ǫu0 = 0 Monte Carlo
u(m−1)
i
ℓm = 3 u(m)
i
u0 u1 u2 F ′
3
v0
0.5
v1
0.5
v2
0.5
GFFT
(I)FFT GIFFT
Uncover next (unknown) input using outputs and recovered inputs.
Narayanan Rengaswamy & Henry D. Pfister Cyclic Polar Codes June 16, 2015 20 / 32
Decoding Successive Cancellation Decoding
q-ary Symmetric Channel with Erasures (QSCE) with e = 0.5, ǫ = 0. Use Berlekamp-Massey (B-M) and Forney algorithms. ui ∈ F16, vi ∈ F16 ∪ {?}, e = P{vi = u(m)
i
} = 0.5, ǫ = P{vi =?} = 0 eu0 = 0.846, ǫu0 = 0 eu1 = 0.031, ǫu1 = 0.846 Monte Carlo
u(m−1)
i
ℓm = 3 u(m)
i
u0 u1 u2 F ′
3
v0
0.5
v1
0.5
v2
0.5
GFFT
(I)FFT GIFFT
Uncover next (unknown) input using outputs and recovered inputs.
Narayanan Rengaswamy & Henry D. Pfister Cyclic Polar Codes June 16, 2015 20 / 32
Decoding Successive Cancellation Decoding
q-ary Symmetric Channel with Erasures (QSCE) with e = 0.5, ǫ = 0. Use Berlekamp-Massey (B-M) and Forney algorithms. ui ∈ F16, vi ∈ F16 ∪ {?}, e = P{vi = u(m)
i
} = 0.5, ǫ = P{vi =?} = 0 eu0 = 0.846, ǫu0 = 0 eu1 = 0.031, ǫu1 = 0.846 eu2 = 0.055, ǫu2 = 0.452 Monte Carlo
u(m−1)
i
ℓm = 3 u(m)
i
u0 u1 u2 F ′
3
v0
0.5
v1
0.5
v2
0.5
GFFT
(I)FFT GIFFT
Uncover next (unknown) input using outputs and recovered inputs.
Narayanan Rengaswamy & Henry D. Pfister Cyclic Polar Codes June 16, 2015 20 / 32
Decoding Successive Cancellation Decoding
q-ary Symmetric Channel with Erasures (QSCE) with e = 0.5, ǫ = 0. Use Berlekamp-Massey (B-M) and Forney algorithms. ui ∈ F16, vi ∈ F16 ∪ {?}, e = P{vi = u(m)
i
} = 0.5, ǫ = P{vi =?} = 0 Theoretical 1 − (1 − e)3 = 0.875 1 − (1 − e)3 = 0.875 1 − [3(1 − e)2e + (1 − e)3] = 0.5...
u(m−1)
i
ℓm = 3 u(m)
i
u0 u1 u2 F ′
3
v0
0.5
v1
0.5
v2
0.5
GFFT
(I)FFT GIFFT
Uncover next (unknown) input using outputs and recovered inputs.
Narayanan Rengaswamy & Henry D. Pfister Cyclic Polar Codes June 16, 2015 20 / 32
Decoding Successive Cancellation Decoding
q-ary Symmetric Channel with Erasures (QSCE) with e = 0.5, ǫ = 0. Use Berlekamp-Massey (B-M) and Forney algorithms. ui ∈ F256, vi ∈ F256 ∪ {?}, e = P{vi = u(m)
i
} = 0.5, ǫ = P{vi =?} = 0 Monte Carlo eu0 = 0.875, ǫu0 = 0 eu1 = 0.002, ǫu1 = 0.875 eu2 = 0.003, ǫu2 = 0.482 ...
u(m−1)
i
ℓm = 3 u(m)
i
u0 u1 u2 F ′
3
v0
0.5
v1
0.5
v2
0.5
GFFT
(I)FFT GIFFT
Uncover next (unknown) input using outputs and recovered inputs.
Narayanan Rengaswamy & Henry D. Pfister Cyclic Polar Codes June 16, 2015 20 / 32
Decoding Successive Cancellation Decoding
For a length-ℓ block, bounded by Cℓ2 operations for some C > 0. Since there are
j=m ℓj = N/ℓm blocks at stage ℓm, the decoding
complexity is bounded by
n
ℓj
m
n
ℓm ≤ CNn max
m ℓm.
For comparison, complexity of standard polar codes for N = 2n is O(NlogN) under SC decoding.
Narayanan Rengaswamy & Henry D. Pfister Cyclic Polar Codes June 16, 2015 21 / 32
Decoding Successive Cancellation Decoding
Narayanan Rengaswamy & Henry D. Pfister Cyclic Polar Codes June 16, 2015 22 / 32
Results
1
Polar Codes Overview
2
Cyclic Polar Codes Galois field Fourier transform Cyclic polar codes
3
Decoding Successive Cancellation Decoding
4
Results For the q-ary Erasure Channel For the q-ary Symmetric Channel
5
Conclusions
Narayanan Rengaswamy & Henry D. Pfister Cyclic Polar Codes June 16, 2015 23 / 32
Results For the q-ary Erasure Channel
0.4 0.5 0.6 0.7 0.8 0.9 1 10−2 10−1 100 Channel Erasure Rate Block Erasure Rate
Cyclic q = 257, N = 28 Cyclic q = 256, N = 255 Comparison of performance of standard polar and cyclic polar codes. δ = 0.1; ǫ = 0.5 produced rates 0.328 for N = 256, 0.384 for N = 255.
Narayanan Rengaswamy & Henry D. Pfister Cyclic Polar Codes June 16, 2015 24 / 32
Results For the q-ary Symmetric Channel
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 10−2 10−1 100 Channel Error Rate Block Error Rate
(256, 84) RS code (256, 84) CP code (255, 98) RS code (255, 98) CP code (16, 4) RS code (16, 4) CP code
Performance of QEC-designed cyclic polar (CP) codes on the QSC. δ = 0.1; ǫ = 0.5 produced rates 0.328 for N = 256, 0.384 for N = 255, 0.25 for N = 16.
Narayanan Rengaswamy & Henry D. Pfister Cyclic Polar Codes June 16, 2015 25 / 32
Conclusions
1
Polar Codes Overview
2
Cyclic Polar Codes Galois field Fourier transform Cyclic polar codes
3
Decoding Successive Cancellation Decoding
4
Results For the q-ary Erasure Channel For the q-ary Symmetric Channel
5
Conclusions
Narayanan Rengaswamy & Henry D. Pfister Cyclic Polar Codes June 16, 2015 26 / 32
Conclusions
Cyclic Polar (CP) code construction for arbitrary blocklength N.
Narayanan Rengaswamy & Henry D. Pfister Cyclic Polar Codes June 16, 2015 27 / 32
Conclusions
Cyclic Polar (CP) code construction for arbitrary blocklength N. Performance of CP code:
QEC: Outperforms standard polar code under deterministic hard-decision SC decoding with much higher rates and larger polarization kernels.
Narayanan Rengaswamy & Henry D. Pfister Cyclic Polar Codes June 16, 2015 27 / 32
Conclusions
Cyclic Polar (CP) code construction for arbitrary blocklength N. Performance of CP code:
QEC: Outperforms standard polar code under deterministic hard-decision SC decoding with much higher rates and larger polarization kernels. QSC: With the design on QEC, outperforms hard-decision decoding of a similar RS code under soft-decision SC decoding; demonstrated for N = 28.
Narayanan Rengaswamy & Henry D. Pfister Cyclic Polar Codes June 16, 2015 27 / 32
Conclusions
Cyclic Polar (CP) code construction for arbitrary blocklength N. Performance of CP code:
QEC: Outperforms standard polar code under deterministic hard-decision SC decoding with much higher rates and larger polarization kernels. QSC: With the design on QEC, outperforms hard-decision decoding of a similar RS code under soft-decision SC decoding; demonstrated for N = 28. QSC: With the design on QEC, inferior to hard-decision decoding of a similar RS code under algebraic hard-decision SC decoding; demonstrated for N = 255 = 3 · 5 · 17.
Narayanan Rengaswamy & Henry D. Pfister Cyclic Polar Codes June 16, 2015 27 / 32
Conclusions
Theoretical analysis for mixed-size kernels to prove polarization and see if these codes are capacity-achieving on arbitrary q-DMCs. Better decoding strategies for finite length cyclic polar codes (essentially better RS decoders for the small blocks). Effect of ordering of the factors of blocklength N in the code graph. Performance under random and burst errors/erasures. Optimizing computational complexity for q-ary alphabets. ... and much more!
Narayanan Rengaswamy & Henry D. Pfister Cyclic Polar Codes June 16, 2015 28 / 32
Conclusions
Narayanan Rengaswamy & Henry D. Pfister Cyclic Polar Codes June 16, 2015 29 / 32
Conclusions
Narayanan Rengaswamy & Henry D. Pfister Cyclic Polar Codes June 16, 2015 30 / 32
Conclusions
Cyclic codes are (principal) ideals in the ring of polynomials GF(q)[x]/(xn − 1). Find a primitive Nth root of unity ω in GF(q). The cyclic subgroup generated by ω gives all the N roots of the equation xN − 1 = 0. Hence we can factorize as xN − 1 =
N−1
(x − ωi) Our generator polynomial g(x) is the product of a subset of these factors and thus divides xN − 1. Since g(x) is monic and is the minimal polynomial of degree N − k in GF(q)[x]/(xn − 1), it is a generator for the ideal. Thus the code is cyclic with generator g(x).
Narayanan Rengaswamy & Henry D. Pfister Cyclic Polar Codes June 16, 2015 31 / 32
Conclusions
The capacity of q-ary symmetric channel with erasures (QSCE) with parameters α and β representing the probability of symbol erasure and symbol error, respectively, is
C = (1 − α) + (1 − α)logq 1 − α − β 1 − α
1 − α − β β
Hence, the capacities of q-ary erasure channel (QEC) and q-ary symmetric channel (QSC) are CQEC = 1 − α and CQSC = 1 − Hq(β) where, Hq(·) is the q-ary entropy function.
Narayanan Rengaswamy & Henry D. Pfister Cyclic Polar Codes June 16, 2015 32 / 32