A General Procedure to Design Good Codes at a Target BER

A General Procedure to Design Good odes at a Target BER Speaker: Xiao Ma 1 maxiao@mail.sysu.edu.cn Joint work with: hulong Liang 1, Qiutao Zhuang 1, and Baoming Bai 2 1 Dept. Electronics and omm. Eng., Sun Yat-sen University, Guangzhou 510006, GD, hina 2 The State Key Lab of ISN, Xidian University, Xi an 710071, SN, hina IST, Bremen, August 19, 2014. Liang, X. Ma, Q. Zhuang, and B. Bai (SYSU) A General Procedure to Design Good odes Bremen, August 19, 2014 1 / 55

Outline 1 Review of BMST Repetition Increases Reliability Superposition Increases Efficiency 2 A General Procedure of Designing BMST 3 Genie-aided Bounds 4 Two-phase Decoding 5 Simulation Results 6 onclusions and Future Works. Liang, X. Ma, Q. Zhuang, and B. Bai (SYSU) A General Procedure to Design Good odes Bremen, August 19, 2014 2 / 55

Repetition Increases Reliability 10 0 u Enc v m c (0) w (1) w (m) BI- BI- BI- y (0) y (1) y (m) BER 10 1 10 2 10 3 10 4 10 5 once 10 6 0 1 2 3 4 5 6 7 8 9 SNR (db) The codeword is transmitted once. We assume a basic code [n,k], whose performance curve in terms of BER versus SNR is available. In this talk, we assume that [N,K ] B, the B-fold artesian product of a short block code [N,K ], consisting of all vectors of the form (v 0,v 1,,v B 1 ), where each v i is a codeword in the [N,K ] code for 0 i B 1.. Liang, X. Ma, Q. Zhuang, and B. Bai (SYSU) A General Procedure to Design Good odes Bremen, August 19, 2014 4 / 55

Repetition Increases Reliability 10 0 u Enc v m c (0) c (1) c (m) BI- BI- BI- y (0) y (1) y (m) BER 10 1 10 2 10 3 10 4 10 5 twice 10log 10 (2) once 10 6 0 1 2 3 4 5 6 7 8 9 SNR (db) The same codeword is transmitted twice. The performance curve shifts to the left by 3 db.. Liang, X. Ma, Q. Zhuang, and B. Bai (SYSU) A General Procedure to Design Good odes Bremen, August 19, 2014 5 / 55

Repetition Increases Reliability 10 0 u Enc v m c (0) c (1) c (m) BI- BI- BI- y (0) y (1) y (m) BER 10 1 10 2 10 3 10 4 10 5 m1 times twice 10log 10 (2) once 10log 10 (m1) 10 6 0 1 2 3 4 5 6 7 8 9 SNR (db) The same codeword is transmitted m1 times. The performance curve shifts to the left by 10log 10 (m1) db. Repetition increases reliability but decreases efficiency (code rate).. Liang, X. Ma, Q. Zhuang, and B. Bai (SYSU) A General Procedure to Design Good odes Bremen, August 19, 2014 6 / 55

Superposition Increases Efficiency 10 0 u u Enc u Enc v Enc w w v v w w w c BI- y w c BI- y c BI- y BER 10 1 10 2 10 3 10 4 10 5 once 10 6 0 1 2 3 4 5 6 7 8 9 SNR (db) The first transmission Initially, the transmitter sends a codeword from the code that corresponds to the first data block.. Liang, X. Ma, Q. Zhuang, and B. Bai (SYSU) A General Procedure to Design Good odes Bremen, August 19, 2014 7 / 55

Superposition Increases Efficiency 10 0 u u Enc u Enc Enc v v w w v w c BI- y w w c BI- y w c BI- y BER 10 1 10 2 10 3 10 4 10 5 10log 10 (2) twice once 10 6 0 1 2 3 4 5 6 7 8 9 SNR (db) The second transmission Since the short code is weak, the receiver is unable to recover reliably the data from the current received block. Hence the transmitter transmits the codeword (possibly in its interleaved version) one more time.. Liang, X. Ma, Q. Zhuang, and B. Bai (SYSU) A General Procedure to Design Good odes Bremen, August 19, 2014 8 / 55

Superposition Increases Efficiency 10 0 u u Enc u Enc Enc v v w w v w c BI- y w w c BI- y w c BI- y BER 10 1 10 2 10 3 10 4 10 5 BMST 10log 10 (2) twice once 10 6 0 1 2 3 4 5 6 7 8 9 SNR (db) The second transmission Since the short code is weak, the receiver is unable to recover reliably the data from the current received block. Hence the transmitter transmits the codeword (possibly in its interleaved version) one more time. In the meanwhile, a fresh codeword from that corresponds to the second data block is superimposed on the second block transmission.. Liang, X. Ma, Q. Zhuang, and B. Bai (SYSU) A General Procedure to Design Good odes Bremen, August 19, 2014 9 / 55

Superposition Increases Efficiency u # u Enc Enc v v # # w!#"# u!$ Enc v $ w!$" w!#% # w!"# c w!% c # c $ BI- y BI- BI- y # y $ BER 10 0 10 1 10 2 10 3 10 4 10 5 BMST 10log 10 (2) twice once 10 6 0 1 2 3 4 5 6 7 8 9 SNR (db) The third transmission for encoding memory m 1 In the third transmission, the current codeword v (2) is superimposed to ( mixed into ) the previous codeword v (1) and then transmitted. This system can be iteratively decoded by passing extrinsic messages between adjacent layers. The performance is intuitively lower bounded by the repetition system.. Liang, X. Ma, Q. Zhuang, and B. Bai (SYSU) A General Procedure to Design Good odes Bremen, August 19, 2014 10 / 55

Superposition Increases Efficiency 10 0 u &( u &'( Enc Enc w )*,.( u ),( Enc v &,( v &'( v &(,-, w ),*'( w )'*,( w )/'( w )'*( Transmission with memory m c &'( BI- y &'( w )'/'( c &( BI- y &( c &,( BI- y &,( BER 10 1 10 2 10 3 10 4 10 5 BMST twice BMST m1 times 10log 10 (2) once 10log (m1) 10 10 6 0 1 2 3 4 5 6 7 8 9 SNR (db) Generally, for a BMST system with memory m, the t-th transmission is a superposition of the current codeword and the m consecutive past codewords, all in their randomly-interleaved version. The code rate remains almost the same, except that termination is needed, while the minimum distance increases very likely by m times for large B m. Hence the error floor can be predicted by shifting the performance curve to the left by 10log 10 (m1) db.. Liang, X. Ma, Q. Zhuang, and B. Bai (SYSU) A General Procedure to Design Good odes Bremen, August 19, 2014 11 / 55

Summary The BMST system The encoding diagram of a BMST system with memory m. c (t) m-1 m u (t) EN D D D v (t) v (t-1) v (t-2) v (t-m1) v (t-m) The normal graph for a BMST system with L4 and m 2. (0) (1) (2) (3) (4) (5) 1 2 1 2 1 2 0 0 0 0 1 2 V (0) V (1) V (2) V (3) a decoding layer U (0) U (1) U (2) U (3). Liang, X. Ma, Q. Zhuang, and B. Bai (SYSU) A General Procedure to Design Good odes Bremen, August 19, 2014 12 / 55

Summary The BMST system The encoding diagram of a BMST system with memory m. c (t) m-1 m u (t) EN D D D v (t) v (t-1) v (t-2) v (t-m1) v (t-m) The normal graph for a BMST system with L4 and m 2. (0) (1) (2) (3) (4) (5) 1 2 1 2 1 2 0 0 0 0 1 2 V (0) V (1) V (2) V (3) a decoding layer U (0) U (1) U (2) The lower bound can be obtained by shifting the performance curve of the basic code to the left by 10log 10 (m1) db. U (3). Liang, X. Ma, Q. Zhuang, and B. Bai (SYSU) A General Procedure to Design Good odes Bremen, August 19, 2014 12 / 55

Outline 1 Review of BMST 2 A General Procedure of Designing BMST 3 Genie-aided Bounds 4 Two-phase Decoding 5 Simulation Results 6 onclusions and Future Works. Liang, X. Ma, Q. Zhuang, and B. Bai (SYSU) A General Procedure to Design Good odes Bremen, August 19, 2014 13 / 55

A General Procedure of Designing BMST With the genie-aided lower bound, to construct a BMST system of a given rate R with a target BER of p target, we can perform the following steps. 1 Take a code [N,K ] B with the given rate R as the basic code. In order to approach the channel capacity, we set the code length n NB 10000 in our simulations; 2 Find the performance curve f basic (γ b ) of the basic code. From this curve, find the required SNR ( 1 ) to achieve the target BER. That is, findγ σ 2 target such that f basic (γ target ) p target ; 3 Find the Shannon limit for the code rate, denoted byγ lim ; 4 Determine the encoding memory by 10log 10 (m1) γ target γ lim. That is, m 10 γ target γ lim 10 1, (1) where x stands for the integer that is closest to x. 5 Generate m 1 interleavers randomly.. Liang, X. Ma, Q. Zhuang, and B. Bai (SYSU) A General Procedure to Design Good odes Bremen, August 19, 2014 14 / 55

onstruction Examples BMST with Different ode Rates Table: The encoding memories required to approach the corresponding Shannon limits using BMST systems for different code rates at given target BERs Basic codes p target γ target (db) γ lim (db) γ target γ lim (db) m R [8,1] 1250 10 3 0.77 7.23 8.00 6 R [8,1] 1250 10 6 4.51 7.23 11.74 14 R [4,1] 2500 10 3 3.78 3.80 7.58 5 R [4,1] 2500 10 6 7.52 3.80 11.32 13 R [2,1] 5000 10 3 6.79 0.19 6.60 4 R [2,1] 5000 10 5 9.59 0.19 9.40 8 R [2,1] 5000 10 6 10.53 0.19 10.34 10 R [2,1] 5000 10 15 14.99 0.19 14.80 30 SP [4,3] 2500 10 3 7.62 3.39 4.23 2 SP [4,3] 2500 10 6 10.91 3.39 7.52 5 SP [8,7] 1250 10 3 7.51 4.60 2.91 1 SP [8,7] 1250 10 6 11.20 5.27 5.93 3. Liang, X. Ma, Q. Zhuang, and B. Bai, Spatial coupling of generator matrix: A general approach to design of good codes at a target BER, submitted to IEEE Trans. ommun.. [Online]. Available: http://arxiv.org/abs/1405.2524.. Liang, X. Ma, Q. Zhuang, and B. Bai (SYSU) A General Procedure to Design Good odes Bremen, August 19, 2014 15 / 55

A onstruction Example BMST with rate-1/2 10 0 10 1 R[2,1] 10 2 BER 10 3 10 4 10 5 10 6 10 7 0 1 2 3 4 5 6 7 8 9 10 11 SNR (db) Figure: Performance of the BMST systems with the R [2,1] 5000 as the basic code. The target BERs are 10 3 and 10 6. The systems encode L100000 sub-blocks of data and decode with the SWD algorithm of a maximum iteration I max 18.. Liang, X. Ma, Q. Zhuang, and B. Bai (SYSU) A General Procedure to Design Good odes Bremen, August 19, 2014 16 / 55

A onstruction Example BMST with rate-1/2 10 0 10 1 R[2,1] Shannon limit of rate 1/2 10 2 BER 10 3 10 4 10 5 10 6 10 7 0 1 2 3 4 5 6 7 8 9 10 11 SNR (db) Figure: Performance of the BMST systems with the R [2,1] 5000 as the basic code. The target BERs are 10 3 and 10 6. The systems encode L100000 sub-blocks of data and decode with the SWD algorithm of a maximum iteration I max 18.. Liang, X. Ma, Q. Zhuang, and B. Bai (SYSU) A General Procedure to Design Good odes Bremen, August 19, 2014 17 / 55

A onstruction Example BMST with rate-1/2 10 0 10 1 R[2,1] Shannon limit of rate 1/2 10 2 BER 10 3 10 4 6.60dB 10 5 10 6 10 7 0 1 2 3 4 5 6 7 8 9 10 11 SNR (db) Figure: Performance of the BMST systems with the R [2,1] 5000 as the basic code. The target BERs are 10 3 and 10 6. The systems encode L100000 sub-blocks of data and decode with the SWD algorithm of a maximum iteration I max 18.. Liang, X. Ma, Q. Zhuang, and B. Bai (SYSU) A General Procedure to Design Good odes Bremen, August 19, 2014 18 / 55

A onstruction Example BMST with rate-1/2 10 0 10 1 R[2,1] Shannon limit of rate 1/2 Lower bound for m 4 10 2 BER 10 3 10 4 6.60dB 10log 10 (41)dB 10 5 10 6 10 7 0 1 2 3 4 5 6 7 8 9 10 11 SNR (db) Figure: Performance of the BMST systems with the R [2,1] 5000 as the basic code. The target BERs are 10 3 and 10 6. The systems encode L100000 sub-blocks of data and decode with the SWD algorithm of a maximum iteration I max 18.. Liang, X. Ma, Q. Zhuang, and B. Bai (SYSU) A General Procedure to Design Good odes Bremen, August 19, 2014 19 / 55

A onstruction Example BMST with rate-1/2 10 0 10 1 10 2 R[2,1] Shannon limit of rate 1/2 Lower bound for m 4 BMST, m 4, d 12, p target 10 3 BER 10 3 10 4 6.60dB 10log 10 (41)dB 10 5 10 6 10 7 0 1 2 3 4 5 6 7 8 9 10 11 SNR (db) Figure: Performance of the BMST systems with the R [2,1] 5000 as the basic code. The target BERs are 10 3 and 10 6. The systems encode L100000 sub-blocks of data and decode with the SWD algorithm of a maximum iteration I max 18.. Liang, X. Ma, Q. Zhuang, and B. Bai (SYSU) A General Procedure to Design Good odes Bremen, August 19, 2014 20 / 55

A onstruction Example BMST with rate-1/2 10 0 10 1 10 2 R[2,1] Shannon limit of rate 1/2 Lower bound for m 4 BMST, m 4, d 12, p target 10 3 BER 10 3 10 4 6.60dB 10log 10 (41)dB 10 5 10 6 10.34dB 10 7 0 1 2 3 4 5 6 7 8 9 10 11 SNR (db) Figure: Performance of the BMST systems with the R [2,1] 5000 as the basic code. The target BERs are 10 3 and 10 6. The systems encode L100000 sub-blocks of data and decode with the SWD algorithm of a maximum iteration I max 18.. Liang, X. Ma, Q. Zhuang, and B. Bai (SYSU) A General Procedure to Design Good odes Bremen, August 19, 2014 21 / 55

A onstruction Example BMST with rate-1/2 10 0 10 1 10 2 R[2,1] Shannon limit of rate 1/2 Lower bound for m 4 BMST, m 4, d 12, p target 10 3 Lower bound for m 10 BER 10 3 10 4 6.60dB 10log 10 (41)dB 10 5 10 6 10 7 10.34dB 10log 10 (101)dB 0 1 2 3 4 5 6 7 8 9 10 11 SNR (db) Figure: Performance of the BMST systems with the R [2,1] 5000 as the basic code. The target BERs are 10 3 and 10 6. The systems encode L100000 sub-blocks of data and decode with the SWD algorithm of a maximum iteration I max 18.. Liang, X. Ma, Q. Zhuang, and B. Bai (SYSU) A General Procedure to Design Good odes Bremen, August 19, 2014 22 / 55

A onstruction Example BMST with rate-1/2 BER 10 0 10 1 10 2 10 3 10 4 R[2,1] Shannon limit of rate 1/2 Lower bound for m 4 BMST, m 4, d 12, p target 10 3 Lower bound for m 10 BMST, m 10, d 30, p 10 6 target 6.60dB 10log 10 (41)dB 10 5 10 6 10 7 10.34dB 10log 10 (101)dB 0 1 2 3 4 5 6 7 8 9 10 11 SNR (db) Figure: Performance of the BMST systems with the R [2,1] 5000 as the basic code. The target BERs are 10 3 and 10 6. The systems encode L100000 sub-blocks of data and decode with the SWD algorithm of a maximum iteration I max 18.. Liang, X. Ma, Q. Zhuang, and B. Bai (SYSU) A General Procedure to Design Good odes Bremen, August 19, 2014 23 / 55

onstruction Examples BMST with Different ode Rates 10 0 10 1 10 2 10 3 Shannon limit of rate 1/8 BMST, m 6, d 18, p target 10 3 BMST, m 14, d 42, p target 10 6 Lower bound for m 6 Lower bound for m 14 BER 10 4 10 5 10 6 10 7 10 8 8 7.5 7 6.5 6 5.5 5 4.5 4 3.5 3 SNR (db) Figure: Performance of the BMST systems with the R [8,1] 1250 as the basic code. The target BERs are 10 3 and 10 6. The systems encode L100000 sub-blocks of data and decode with the SWD algorithm of a maximum iteration I max 18.. Liang, X. Ma, Q. Zhuang, and B. Bai (SYSU) A General Procedure to Design Good odes Bremen, August 19, 2014 24 / 55

onstruction Examples BMST with Different ode Rates 10 0 10 1 10 2 10 3 Shannon limit of rate 1/4 BMST, m 5, d 15, p target 10 3 BMST, m 13, d 39, p target 10 6 Lower bound for m 5 Lower bound for m 13 BER 10 4 10 5 10 6 10 7 10 8 4 3.5 3 2.5 2 1.5 1 0.5 0 SNR (db) Figure: Performance of the BMST systems with the R [4,1] 2500 as the basic code. The target BERs are 10 3 and 10 6. The systems encode L100000 sub-blocks of data and decode with the SWD algorithm of a maximum iteration I max 18.. Liang, X. Ma, Q. Zhuang, and B. Bai (SYSU) A General Procedure to Design Good odes Bremen, August 19, 2014 25 / 55

onstruction Examples BMST with Different ode Rates 10 0 10 1 10 2 10 3 Shannon limit of rate 3/4 BMST, m 2, d 6, p target 10 3 BMST, m 5, d 15, p target 10 6 Lower bound for m 2 Lower bound for m 5 BER 10 4 10 5 10 6 10 7 10 8 3 3.5 4 4.5 5 5.5 6 6.5 7 7.5 8 SNR (db) Figure: Performance of the BMST systems with the SP [4,3] 2500 as the basic code. The target BERs are 10 3 and 10 6. The systems encode L100000 sub-blocks of data and decode with the SWD algorithm of a maximum iteration I max 18.. Liang, X. Ma, Q. Zhuang, and B. Bai (SYSU) A General Procedure to Design Good odes Bremen, August 19, 2014 26 / 55

onstruction Examples BMST with Different ode Rates 10 0 10 1 10 2 10 3 Shannon limit of rate 7/8 BMST, m 1, d 3, p target 10 3 BMST, m 3, d 9, p target 10 6 Lower bound for m 2 Lower bound for m 3 BER 10 4 10 5 10 6 10 7 10 8 5 5.5 6 6.5 7 7.5 8 8.5 9 9.5 10 SNR (db) Figure: Performance of the BMST systems with the SP [8,7] 1250 as the basic code. The target BERs are 10 3 and 10 6. The systems encode L100000 sub-blocks of data and decode with the SWD algorithm of a maximum iteration I max 18.. Liang, X. Ma, Q. Zhuang, and B. Bai (SYSU) A General Procedure to Design Good odes Bremen, August 19, 2014 27 / 55

onstruction Examples BMST with Different ode Rates Rate (bits/channel use) 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 Performance of the BMST system apacity bound with BPSK signalling 0 11 10 9 8 7 6 5 4 3 2 1 0 1 2 3 4 5 6 7 8 SNR (db) Figure: The required SNRs (1/σ 2 ) for the BMST system using repetition codes and single-parity-check codes to achieve the BER of 10 6 over the BI-.. Liang, X. Ma, Q. Zhuang, and B. Bai (SYSU) A General Procedure to Design Good odes Bremen, August 19, 2014 28 / 55

onstruction Examples Hadamard Transform oset odes u 8 u 9:; 0 >?@A BDEFG HIJK 0 Π < H M T 8 T; T L Table: The Memory Required for Each ode Rate Using the BMST of HT-coset odes with N 8 to Approach the Shannon Limit at the BER of 10 5 Rate R K/8 1/8 2/8 3/8 4/8 5/8 6/8 7/8 γ K (db) -1.2-0.8-0.3 0.2 0.8 1.6 2.9 γ K (db) 9.6 9.8 8.4 7.7 8.9 8.6 8.2 Gapγ K γ K (db) 10.8 10.6 8.7 7.5 8.1 7.0 5.3 Memory m K 11 10 6 5 5 4 2 v 0 v 1 v 2 v 3 v 4 v 5 v 6 v 7. Liang, J.Hu, X. Ma, and B. Bai, A new class of multiple-rate codes based on block markov superposition transmission, submitted to IEEE Trans. Signal Process.. [Online]. Available: http://arxiv.org/abs/1308.4809.. Liang, X. Ma, Q. Zhuang, and B. Bai (SYSU) A General Procedure to Design Good odes Bremen, August 19, 2014 29 / 55

BMST-HT Systems Rate (bits/channel use) 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 Performance of the BMST HT system apacity bound with BPSK signalling 0 11 10 9 8 7 6 5 4 3 2 1 0 1 2 3 4 5 6 7 8 SNR (db) Figure: The required SNRs (1/σ 2 ) for the BMST-HT system using the artesian products of HT-coset codes [8,K ] 1250 (1 K 7) to achieve the BER of 10 5 over the BI-.. Liang, X. Ma, Q. Zhuang, and B. Bai (SYSU) A General Procedure to Design Good odes Bremen, August 19, 2014 30 / 55

BMST-HT Systems Rate (bits/channel use) 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 Performance of the BMST HT system apacity bound with BPSK signalling 0 11 10 9 8 7 6 5 4 3 2 1 0 1 2 3 4 5 6 7 8 SNR (db) Figure: The required SNR for the BMST-HT system using the artesian products of HT-coset codes [16,K ] 625 (1 K 15) to achieve the BER of 10 5 over the BI-.. Liang, X. Ma, Q. Zhuang, and B. Bai (SYSU) A General Procedure to Design Good odes Bremen, August 19, 2014 31 / 55

onstructing BMST Systems with Extremely Low Error Floor From the previous examples, we can see that good codes can be constructed following the general procedure when the the target BER is around 10 5. An unavoidable question is that whether or not this procedure is applicable to the case when the target BER is extremely low. The difficulty lies in the fact that it is very time-consuming and even infeasible to verify this matchness by conventional software simulation in the extremely low error rate region. We propose a two-phase decoding (TPD) algorithm for the BMST system, whose performance is predictable.. Liang, X. Ma, Q. Zhuang, and B. Bai (SYSU) A General Procedure to Design Good odes Bremen, August 19, 2014 32 / 55

onstructing BMST Systems with Extremely Low Error Floor From the previous examples, we can see that good codes can be constructed following the general procedure when the the target BER is around 10 5. An unavoidable question is that whether or not this procedure is applicable to the case when the target BER is extremely low. The difficulty lies in the fact that it is very time-consuming and even infeasible to verify this matchness by conventional software simulation in the extremely low error rate region. We propose a two-phase decoding (TPD) algorithm for the BMST system, whose performance is predictable. For doing so, we propose a genie-aided decoder and a corresponding genie-aided bound, both of which are useful not only for developing the TPD algorithm but also for predicting the performance of the TPD algorithm.. Liang, X. Ma, Q. Zhuang, and B. Bai (SYSU) A General Procedure to Design Good odes Bremen, August 19, 2014 32 / 55

Genie-aided Bounds Figure: The BMST system. The Genie-aided Lower Bound w [Ya]\_b]Z w [^\_Z w dye_\^z u NOP Enc v NOP c NOP BI- y NOP w QOWOP T S w QORSP u NSP BI- y NSP Enc v NSP c NSP TUS w QSWOP w QSRTVSP w QORTP u QTP c NTP BI- y NTP Enc v NTP w QTROP w [Yb]c]Z w [Yb_c_Z u XYZ Enc v XYZ c XYZ BI- y XYZ w [Yc^Z _ ] w [Ya]c^Z w [Y\]Z w dyb_e]f_z c [Ya]Z BI- y [Y`]Z c [Ya_Z BI- y [Ya_Z Figure: The genie-aided lower bound system. In the derivation of the genie-aided lower bound [Ma], the transmitted data u ( u (0),,u (t 1),u (t1),,u (L 1)) are assumed to be known at the decoder. This is equivalent to assuming a genie who tells the decoder all but one the intermediate codewords v ( v (0),,v (t 1),v (t1),,v (L 1)). This is also equivalent to assuming that w { w (t,i) v (t ) Π i : m t Lm 1,t t,0 i m} are available at the decoder. [Ma] X. Ma,. Liang, K. Huang, and Q. Zhuang, Block Markov superposition transmission: onstruction of big convolutional codes from short codes, revised to IEEE Trans. Inf. Theory. [Online]. Available: http://arxiv.org/abs/1308.4809. Liang, X. Ma, Q. Zhuang, and B. Bai (SYSU) A General Procedure to Design Good odes Bremen, August 19, 2014 34 / 55

Genie-aided Bounds u ghi Enc v ghi n jhplrmi w w jhqlrli w jhrmi l w jhkli thqlunqli w jhqnrni w c ghi BI- y ghi c jhpli BI- y jholi u vwx Enc v vwx } yw{x c { w ywz{x ywx c w ywz x c vwx BI- y vwx c yw{x BI- y yw~{x jhplknqli w jhsnkmi w w jmkni c jhpni BI- y jhpni yw }x c w y z}x Figure: The genie-aided bound system. c yw}x BI- y yw}x The genie-aided assumption is relaxed by assuming a genie who tells the decoder w but with each digit being flipped independently with a probability p genie. The genie observes a noisy version of w through a binary symmetric channel (BS) with crossover probability p genie, denoted by w { w (t,i) : m t Lm 1,t t,0 i m }.. Liang, X. Ma, Q. Zhuang, and B. Bai (SYSU) A General Procedure to Design Good odes Bremen, August 19, 2014 35 / 55

Genie-aided Bounds The Genie-aided Decoding Algorithm At time t, upon the observation of w, we can perform a sub-optimal decoding algorithm to recover u (t) by not taking into account the parameter p genie. ancelation: To remove the effect of w from y (t),,y (tm) as if it were w, we compute m c (ti) w (ti l,l) and ỹ (ti) ( 1) c (ti) y (ti) (2) l0,l i for 0 i m, where denotes the component-wise multiplication between two vectors. Minimization: Find û (t) (equivalently ˆv (t) ) that minimizes m ỹ (ti) ( 1) ˆv (t) Π i 2, (3) i0 where 2 represents the squared Euclidean norm. Output: Output û (t) as the decoding result.. Liang, X. Ma, Q. Zhuang, and B. Bai (SYSU) A General Procedure to Design Good odes Bremen, August 19, 2014 36 / 55

Genie-aided Bounds u ƒ Enc v ƒ ˆ Š ƒ c w ƒ ƒ c w ƒ c ƒ BI- y ƒ c Š ƒ BI- y ƒ u Enc v c (0) c (1) BS p Œ Ž BS p Œ Ž BI- BI- y (0) y (1) ˆƒ c w ˆƒ c Šˆƒ BI- y Šˆƒ m c (m) BS p Œ Ž BI- Figure: The genie-aided system (left) and its equivalent system (right). To derive the genie-aided bounds, we need the following two assumptions. Assumption 1. Each component w (t,i) j of w is statistically independent of the corresponding received signal y (ti) j. Assumption 2. The components of w are statistically independent. With the above two assumptions, we can see that each bit v (t) j is transmitted m1 times through a BI- with a flipping error p flip, which is given by ( m r p flip )p ( ) genie 1 p m r genie 1 ( ) 1 2p m genie. (4) r 2 r is odd y (m). Liang, X. Ma, Q. Zhuang, and B. Bai (SYSU) A General Procedure to Design Good odes Bremen, August 19, 2014 37 / 55

Genie-aided Bounds Assume that u (t) 0 is transmitted. Let the decoding output û (t) 0 corresponding to the codeword ˆv (t) with Hamming weight h. Then the pairwise error probability Pr { u (t) û (t)} is the same as the probability m Pr ỹ (ti) ( 1) w (t,i) 2 m ỹ (ti) ( 1)ŵ (t,i) 2. (5) i0 It is not surprising ( that the pair-wise error probability depends only on the Hamming weight W H (ŵ (t,0),,ŵ (t,m) ) ) (m1)h. Since each ỹ (t) j is distributed according to N ( 1,σ 2) with probability p flip andn ( 1,σ 2) with probability 1 p flip, we have Pr { u (t) û (t)} r0 i0 (m1)h ( ) ( ) (m1)h p r ) (m1)h r (m1)h 2r r flip( 1 pflip Q. (6) (m1)hσ Using the union bound, the genie-aided bound is given by f genie (γ b ) K N g1 h1 g K A g,h Pr { u (t) û (t)}, (7) where A g,h, 1 g K,1 h N, are coefficients of the IOWEF of the basic code.. Liang, X. Ma, Q. Zhuang, and B. Bai (SYSU) A General Procedure to Design Good odes Bremen, August 19, 2014 38 / 55

Genie-aided Bounds ẅ ẅ z (0) u v y (0) Enc ẅ ẅ z (1) y (1) ẅ ẅ z (30) y (30) BER 10 0 10 1 10 2 10 3 10 4 10 5 10 6 10 7 10 8 10 9 10 10 10 11 10 12 10 13 10 14 10 15 10 16 10 17 10log 10 (31) db Shannon limit of rate 1/2 R[2,1], m 0 p genie 10 1, p flip 5.0 10 1 p genie 10 2, p flip 2.3 10 1 p genie 10 3, p flip 2.9 10 2 p genie 10 4, p flip 3.0 10 3 p genie 10 5, p flip 3.0 10 4 p genie 10 6, p flip 3.0 10 5 p genie 10 7, p flip 3.0 10 6 p genie 0, p flip 0.0 10 18 1 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 SNR (db) Figure: The genie-aided bounds with different p genie for the BMST system with an encoding memory m 30 using R [2,1] B as the basic code. The performance curve for the basic code (m 0) is also plotted here.. Liang, X. Ma, Q. Zhuang, and B. Bai (SYSU) A General Procedure to Design Good odes Bremen, August 19, 2014 39 / 55

Two-phase Decoding As seen from the previous example, if we could find a genie who tells us w with an error probability p genie 10 5, we can perform the GAD algorithm. If this is the case, the performance in the extremely low error rate region can be predicted with the help of the genie-aided bound. Two-phase Decoding Steps The phase-i decoding performs the sliding-window decoding algorithm, which serves as a genie. The phase-ii decoding performs the genie-aided decoding algorithm, which cleans up the residual errors once when the phase-i decoding lowers the BER down to around 10 5. š œ žÿ Ÿ 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 š œ žÿ Ÿ š œ žÿ Ÿ û û û û û û û û û û Figure: The decoding of the two-phase decoding of a BMST system with m 2,L10, and d 3... Liang, X. Ma, Q. Zhuang, and B. Bai (SYSU) A General Procedure to Design Good odes Bremen, August 19, 2014 41 / 55

Two-phase Decoding Message flow tree (0) 0 1 2 0 1 2 0 1 2 0 V (0) U (0) V (1) (1) U (1) V (2) (2) U (2) V (3) (3) U (3) 1 2 Figure: The message flow in the normal graph. (4) (5) a decoding layer º ¼ ³» ² ½ ª ««± µ ¹ Figure: A message-flow neighborhood of depth 1. In this figure, m 2 and N 3.. Liang, X. Ma, Q. Zhuang, and B. Bai (SYSU) A General Procedure to Design Good odes Bremen, August 19, 2014 42 / 55

Two-phase Decoding Message flow tree (0) 0 1 2 0 1 2 0 1 2 0 V (0) U (0) V (1) (1) U (1) V (2) (2) U (2) V (3) (3) U (3) 1 2 Figure: The normal graph for a BMST system with L4 and m 2. (4) (5) a decoding layer ¾ À ÁÂÃ Ä Ä ¾ ¾ ¾ ¾ Figure: A message-flow neighborhood of depth 1. In this figure, m 2 and N 3. Å ¾ ¾. Liang, X. Ma, Q. Zhuang, and B. Bai (SYSU) A General Procedure to Design Good odes Bremen, August 19, 2014 43 / 55

Two-phase Decoding Message flow tree (0) 0 1 2 0 1 2 0 1 2 0 V (0) U (0) V (1) (1) U (1) V (2) (2) U (2) V (3) (3) U (3) 1 2 Figure: The normal graph for a BMST system with L4 and m 2. (4) (5) a decoding layer Ç Ì Æ Ç È ÉÊË Ç Æ Æ Æ Æ Figure: A message-flow neighborhood of depth 1. In this figure, m 2 and N 3. Í Ì Æ Æ. Liang, X. Ma, Q. Zhuang, and B. Bai (SYSU) A General Procedure to Design Good odes Bremen, August 19, 2014 44 / 55

Two-phase Decoding Message Flow tree (0) 0 1 2 0 1 2 0 1 2 0 V (0) U (0) V (1) (1) U (1) V (2) (2) U (2) V (3) (3) U (3) 1 2 Figure: The normal graph for a BMST system with L4 and m 2. (4) (5) a decoding layer Ï Ô Î Ï Ð ÑÒÓ Ï Î Î Î Î Figure: A message-flow neighborhood of depth 1. In this figure, m 2 and N 3. Õ Ô Î Î. Liang, X. Ma, Q. Zhuang, and B. Bai (SYSU) A General Procedure to Design Good odes Bremen, August 19, 2014 45 / 55

Two-phase Decoding Message flow tree (0) 0 1 2 0 1 2 0 1 2 0 V (0) U (0) V (1) (1) U (1) V (2) (2) U (2) V (3) (3) U (3) 1 2 Figure: The normal graph for a BMST system with L4 and m 2. (4) (5) a decoding layer Ö Ø ÙÚÛ Ü Ü Ö Ö Ö Ö Figure: A message-flow neighborhood of depth 1. In this figure, m 2 and N 3. Ý Ö Ö. Liang, X. Ma, Q. Zhuang, and B. Bai (SYSU) A General Procedure to Design Good odes Bremen, August 19, 2014 46 / 55

Two-phase Decoding Message flow tree (0) 0 1 2 0 1 2 0 1 2 0 V (0) U (0) V (1) (1) U (1) V (2) (2) U (2) V (3) (3) U (3) 1 2 Figure: The normal graph for a BMST system with L4 and m 2. (4) (5) a decoding layer ß ä Þ ß à áâã ß Þ Þ Þ Þ Figure: A message-flow neighborhood of depth 1. In this figure, m 2 and N 3. å ä Þ Þ. Liang, X. Ma, Q. Zhuang, and B. Bai (SYSU) A General Procedure to Design Good odes Bremen, August 19, 2014 47 / 55

Two-phase Decoding Message flow forest To evaluate the performance, we only need to see what happens to one sub-codeword of [N,K ] B. The codeword is transmitted m1 times, where each transmission is affected by both the AWGN and the flipping error. The statistic independence of these noises is analyzed by considering m(m 1)N trees. ú û üýþ ú ÿ ú ÿ ù ù ù ù ù ú û üýþ ú ÿ ú ÿ ù ù ù ù ù ù ù è æ Pï ë ñ ê ð é ú ò û üýþ ú ÿ ú ÿ ù ù ù ù ù ù ù ( ~ ' è í ç ìæ w ) ú û üýþ ú ÿ ú ÿ ù ù ù ù î óôõö ø óôõö ù ù ú û üýþ óôõö ú ÿ ú ÿ ù ù ù ù ù ù ú û üýþ ú ÿ ú ÿ ù ù ù ù ù ú û üýþ ú ÿ ú ÿ ù ù ù ù ù ù ( ~ ' ) w ù P ú û üýþ ú ÿ ú ÿ ù ù ù ù ù ù ù ú û üýþ ú ÿ ú ÿ ù ù ù ù ù ù ú û üýþ ú ÿ ú ÿ ù ù ù ù ù ù Figure: Diagram of message-flow neighborhoods of depth I for the bit W (t,i) j and the bit W (t,i ). Liang, X. Ma, Q. Zhuang, and B. Bai (SYSU) A General Procedure to Design Good odes Bremen, August 19, 2014. 48 / 55

Two-phase Decoding Summary Assumption 1. Each component w (t,i) j of w is statistically independent of the corresponding received signal y (ti) j. Assumption 2. The components of w are statistically independent. For BMST of [N,K ] B codes, in the case when n m (namely B being large enough for artesian product codes), these two assumptions hold very likely especially for the very beginning several iterations.. Liang, X. Ma, Q. Zhuang, and B. Bai (SYSU) A General Procedure to Design Good odes Bremen, August 19, 2014 49 / 55

Simulation Results 10 0 10 1 10 2 Shannon limit of rate 1/2 R[2,1] 5000, SWD, m 8, d 8 R[2,1] 5000, TPD, p I, m 8, d 8 R[2,1] 5000, TPD, p II, m 8, d 8 Upper bound with p genie p I Lower bound for m 8 10 3 BER 10 4 10 5 10 6 10 7 0 0.5 1 1.5 2 SNR (db) Figure: Performance of the BMST system with the R [2,1] 5000 as the basic code. The target BER is 10 5. The system encodes L100000 sub-blocks of data with the encoding memory m 8 and decodes with a decoding delay d 8 and a maximum iteration I max 18.. Liang, X. Ma, Q. Zhuang, and B. Bai (SYSU) A General Procedure to Design Good odes Bremen, August 19, 2014 51 / 55

Simulation Results BER 10 0 10 1 10 2 10 3 10 4 10 5 10 6 10 7 10 8 10 9 10 10 10 11 10 12 10 13 10 14 10 15 10 16 10 17 Shannon limit of rate 1/2 R[2,1] 5000, SWD, m 30, d 60 R[2,1] 5000, TPD, p I, m 30, d 60 R[2,1] 5000, TPD, p II, m 30, d 60 Upper bound with p genie p I Lower bound for m 30 10 18 0 0.5 1 1.5 2 SNR (db) Figure: Performance of the BMST system with the R [2,1] 5000 as the basic code. The target BER is 10 15. The system encodes L100000 sub-blocks of data with the encoding memory m 30 and decodes with a decoding delay d 60 and a maximum iteration I max 18. At the SNR of 0.5 db, p I 7.2 10 6. Hence, according to the genie-aided bound, p II 4.2 10 17.. Liang, X. Ma, Q. Zhuang, and B. Bai (SYSU) A General Procedure to Design Good odes Bremen, August 19, 2014 52 / 55

onclusions and Future Works onclusions and Future Works We proposed a noisy genie-aided bound for the BMST; We also proposed a two-phase decoding (TPD) algorithm for the BMST, whose performance can be predicted by the noisy genie-aided bound; We presented a general procedure to constuct good BMST codes; The effectiveness of this procedure has been verified for many other channels (not included in this paper); Future work includes studying the similarities and differences between BMST and other types of good codes... Acknowledgements This work was supported by the 973 Program (No. 2012B316100) and the NSF (No. 61172082) of hina.. Liang, X. Ma, Q. Zhuang, and B. Bai (SYSU) A General Procedure to Design Good odes Bremen, August 19, 2014 54 / 55

Thank You for Your Attention!. Liang, X. Ma, Q. Zhuang, and B. Bai (SYSU) A General Procedure to Design Good odes Bremen, August 19, 2014 55 / 55