IEEE C802.16e-04/264. IEEE Broadband Wireless Access Working Group <

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1 I C82.6e-4/264 Project Title Date Submitted I 82.6 roadband Wireless Access Working Group < Irregular Structured DPC Codes Source(s) Victor Stolpman, Jianzhong (Charlie) Zhang, Nico van Waes Nokia 6 Connection Drive Irving, TX 7539 Voice: Fax: victor.stolpman@nokia.com, charlie.zhang@nokia.com, nico.vanwaes@nokia.com e: ecirculation ballot 4c (826-4/5) Abstract In this document, we describe a structured approach to irregular DPC code construction based on seed matrices that are expanded using permutation matrices for purposes of error correction control. These codes have small storage requirements with good block error rate performances over a wide range block sizes. Also described in this document is a structured approach to puncturing irregular DPC codes facilitate rate-compatibility without having to modify the connective net in the encoder or decoder while still offering a wide range of code rates for link optimization. This document, while a full description of DPC code construction by itself, currently contains no specific implementation language, as it is intended primarily as placeholder for ongoing harmonization efforts. Purpose Notice elease Adoption of proposed text as optional feature. This document has been prepared to assist I It is offered as a basis for discussion and is not binding on the contributing individual(s) or organization(s). The material in this document is subject to change in form and content after further study. The contributor(s) reserve(s) the right to add, amend or withdraw material contained herein. The contributor grants a free, irrevocable license to the I to incorporate material contained in this contribution, and any modifications thereof, in the creation of an I Standards publication; to copyright in the I s name any I Standards publication even though it may include portions of this contribution; and at the I s sole discretion to permit others to reproduce in whole or in part the resulting I Standards publication. The contributor also acknowledges and accepts that this contribution may be made public by I 82.6.

2 I C82.6e-4/264 Patent Policy and Procedures The contributor is familiar with the I 82.6 Patent Policy and Procedures < including the statement "I standards may include the known use of patent(s), including patent applications, provided the I receives assurance from the patent holder or applicant with respect to patents essential for compliance with both mandatory and optional portions of the standard." arly disclosure to the Working Group of patent information that might be relevant to the standard is essential to reduce the possibility for delays in the development process and increase the likelihood that the draft publication will be approved for publication. Please notify the Chair <mailto:chair@wirelessman.org> as early as possible, in written or electronic form, if patented technology (or technology under patent application) might be incorporated into a draft standard being developed within the I 82.6 Working Group. The Chair will disclose this notification via the I 82.6 web site <

3 I C82.6e-4/264 Irregular Structured DPC Codes Victor Stolpman, Jianzhong (Charlie) Zhang, Nico van Waes Nokia Introduction: Modern communication systems use Forward rror Correction (FC) codes in an attempt to convey information more reliably through channels with random events. One such FC error control system use ow-density Parity- Check (DPC) codes because of error correcting capabilities that rival the performance of the so-called Turbo- Codes and for their applicability over a wide range of statistical channels [,2,3]. In fact, some random irregular DPC constructions based on edge ensemble designs have error correcting capabilities measured in it rror ate () that are within.5d of the rate distorted Shannon limit for the AWGN channel [3]. Unfortunately, these random PDC code constructions require long codeword constructions (on the order of 6 to 7 bits) in order to achieve these error rates, and despite good performance, these random code constructions often have poor lock rror ate () performances. Hence, these random constructions do not lend themselves well to packet-based communication systems. Yet another disadvantage of random constructions based on edge distribution ensembles is for each codeword length another random construction is needed. Thus, communication systems employing variable block sizes (e.g. TCP/IP) would require multiple code definitions that could consume a significant about of non-volatile memory storage for a large combination of codeword lengths and code rates. An alternative to random DPC construction is structured DPC constructions that rely on a general algorithmic approach to constructing DPC matrices and require much less non-volatile memory than random constructions. Thus, the problem is to design irregular structured DPC codes that have good overall error performance (both and ) for a wide range of code rates and block sizes with attractive storage requirements. The result of such DPC codes is a better performing communication system with lower cost terminals. These factors make such a FC attractive for application over a wide range of products including but not limited to I82.6 and I82.n compliant products. Thus, this exact description below for DPC code construction succeeds at solving the above said problem while out performing competing solutions [4] in overall error performance for similar block sizes and iterations without the non-volatile memory requirements of randomly constructed codes. Seed Matrix, Spreading Matrices and the xpanded DPC Matrix: In this section, we describe and define the binary seed parity-check matrix H SD of dimension ( N SD KSD) NSD) with the code rate defined as K SD NSD. Simply, the seed matrix H SD is a starting matrix used to generate expanded DPC parity-check matrices H through a spreading process. In the Simulation esults section, we specifically define three seed matrices for code rates /2, 2/3, and 3/4 of dimensions ( 26 52), ( 2 6), and ( 56) 4 respectively that solves the earlier stated problem definition. Also it is important to note that these given seed matrices are nearly upper triangular, and as a result all expanded DPC matrices using these seeds in the following described approach will also have a nearly upper triangular construction that lend themselves well for encoding purposes (i.e. near linear encoding complexity with respect to codeword length) [5]. The purpose of these seed matrices is to identify the location and type of sub-matrices in the expanded DPC parity-check matrix H constructed from expanding H with a given set of spreading matrices where all SD 2

4 I C82.6e-4/264 elements of the set are of dimension ( N SPAD N SPAD ). The ones in HSD determine the location of submatrices in the expanded matrix H that contain a spreading matrix from the following set of matrices 2 { P, P, P, P, K P p } SPAD SPAD SPAD SPAD, SPAD where p is a positive prime integer (to be define specifically later), P SPAD denotes the all zeros matrix (i.e. P SPAD = where every matrix element is a zero), P SPAD = I is the identity matrix, P SPAD is a full-rank permutation matrix, we define The zeros in p p P = P P, and so on in like fashion up to P = ( P ). Specifically, 2 SPAD SPAD SPAD SPAD P SPAD here as the single circular-shift permutation matrix, e.g. for N = SPAD 5 P SPAD =. SPAD H SD indicate the location of the sub-matrix P SPAD = in the expanded matrix H. Thus, the ( N K) N where N = N SPAD NSD and K = N SPAD KSD with expanded DPC matrix H is of dimension ( ) sub-matrices consisting of permutation matrices of dimension ( N SPAD N SPAD ) raised to an exponential power from the set of {,,..., p, }, or simply in matrix form where F, j {,,..., p, } exponential matrix is P P H = P F, SPAD F2, SPAD M F( NSD KSD ), SPAD P P P F, 2 SPAD F2,2 SPAD M F( NSD KSD ),2 SPAD i for i,2,,( NSD KSD ) and is of same dimension as the O P P P F, NSD SPAD F2, NSD SPAD M F( NSD KSD ), NSD SPAD = K and j =,2, K, NSD. In matrix form, the final F, F,2 F, N SD F2, F2,2 F2, NSD F = M M O M F( ) ( ) ( ) NSD KSD, F NSD KSD,2 F NSD KSD, NSD H, ( N K ) N ) SD SD SD SD. The selection of the i j set {,,..., p, } uses HSD and is described further in the following section. F, matrix elements of the 3

5 I C82.6e-4/264 xponent Matrix Construction: Specifically, let p be defined as the smallest prime integer that satisfies both then define an interim exponent array matrix as where XPONNT i, j 2,2 3, = M = ( i )( j ) mod p. 2,3 3,2 4, M O 2,4 3,3 4,2 O 2,( NSD KSD ) 3,( NSD KSD ) 4,( NSD KSD 2) M ( NSD KSD + ), O NSD + 2 p and NSPAD p. We 2,( NSD + ) 3, NSD 4,( NSD ) M ( NSD KSD + ),( KSD + 2) Finally, we construct the exponential matrix F used in the expanded DPC matrix H by replacing each one in H with the corresponding matrix element (i.e. same row and column) in the interim exponent matrix SD XPONNT, and we replace each zero in SD when using modulo- p arithmetic in the construction of A Small Construction xample: For example purposes only, let and let H with. Thus, the elements of F belong to the set {,,..., p, } XPONNT. H = SD thus N SD = 6, P SPAD = thus N SPAD = 3, so p = is the smallest prime number that satisfies the specified conditions NSD + 2 p and N p Then, the interim and final exponent matrices as defined above are and the expanded DPC matrix is = XPONNT and F =, 3 8 SPAD. 4

6 I C82.6e-4/264 5 = H. Structured Puncturing: In this section, we introduce a seed puncture-degree sequence SD, d of dimension ( ) [ ] T d d d () () 2 (), SD = d where each element () i d indicates the degree of a variable node corresponding to a codeword element to be punctured in the code defined by SD H. We expand this seed puncture sequence to create an expanded puncture-degree sequence DG p of dimension ( ) ( ) SD N N that contain the variable-degrees corresponding to the columns of the expanded parity-check matrix H of dimension ( ) N K N ) ( = = SD SD 2 SD, DG N N N N T T p p p d p where each element i p indicates the degree of a variable node corresponding to a codeword element to be punctured in the expanded code defined by H and represents the Kronecker product. To offer more flexibility in degree selection, multiple seed sets can be used m SD, d of possible different dimensions ( ) m for m,m,2, = K [ ] T m m m m m d d d ) ( ) ( 2 ) (, SD = d for M m,,2, = K

7 I C82.6e-4/264 corresponding to the degrees to puncture in an intermediate expanded parity-check matrix H m of dimension m ( N K mn where N = m. Although there is no limit on the maximum value of m, it is ( ) SD SD ) SD SPAD desirable to keep storage costs low and hence keep M as a small number (in this report we have used m,2,3,4,5 ). { } p of dimension ( ) ) Then, the expanded puncture-degree sequence N mn DG m SD can be constructed in a similar fashion as above that contains the variable-degrees corresponding to the columns of the expanded paritycheck matrix H of dimension ( N K) N) where N mnsd is a positive integer. p T DG = d = SD [ d d d ] [ ] SD [ d d d d d d d d d ] = m m m = p T SD, m p 2 2 N mn p m m N mnsd N mn m N mnsd where each element p i indicates the degree of a variable node corresponding to a codeword element to be punctured in the expanded code defined by H. The degree-puncture sequence element v i has the degree i p DG is then mapped to variable node-puncture nodes v NOD where each i =,2, K, N mn, p for ( ) m SD p T DG v T NOD = v v 2 v mn mnsd where v i {,2, K, N} for mn i =,2, K, and elements from the set {,2, K, N} occur at most once with in mnsd the variable node puncture sequence v (i.e. cannot puncture the same codeword element twice). Although NOD further optimization can be done in the future, for this document the mapping approach used the very first variable-nodes (starting in the left most column of H and moving right) that have degrees corresponding to order of degree elements in p. DG 6

8 I C82.6e-4/264 In order to achieve a particular effective code rate from a mother code H of dimension (( N K) N), say K = N P we use the very first P elements of FF where P {,,2, K,( m N mnsd) } T NOD for m =,2, K, M, v (i.e. [ v v ] 2 v P ) to puncture code word elements. This nested structure to puncturing, reduces storage complexity and can achieve all the possible code rates, i.e. K K K K K FF,,, K,, N N N 2 K + K provided v NOD is long enough. y puncturing codeword elements, there is no need to change the connective net for multiple code rates given that the effective code rate is within the mother codes capability (i.e. implementation friendly). Of course, there is a useful range of code rates of which outside performance will suffer as with most punctured error correction codes. 7

9 I C82.6e-4/264 8 ate /2 Irregular Seed Parity-Check Matrix: = H SD,53

10 I C82.6e-4/264 9 ate 2/3 Irregular Seed Parity-Check Matrix: = H SD,6

11 I C82.6e-4/264 ate 3/4 Irregular Seed Parity-Check Matrix: H SD,56 =

12 I C82.6e-4/264 Simulation esults: The proposed codes are evaluated and compared with Samsung s structured codes [4]. The results are organized into two subsections. The first subsection includes the results for a minimal set of length n=52, rate r = /2 code, n =728, r=2/3 code and n = 234, r= 3/4 code. The second subsection includes the additional results required for the structured codes, including, r=/2, n=576, 728, 234 codes; r = 2/3, n= 576, 52, 234 codes and r = 3/4, n = 576, 52, 728 codes. All simulations used 5 iterations of the conventional Sum-Product- Algorithm (SPA). Since our current seed matrix sizes are not exact multiples of 48 (this can easily be changed in future version under this construction scheme) our codes do not match the above nominal code rates and block sizes at the same time. Instead, here we present two sets of codes with rates/code word lengths close to these nominal values. The first set matches the code rate requirement, and they are represented by the red curves in the plots. These are designed to match the nominal code rates at the expense of slightly longer or shorter codeword lengths. The other set matches the codeword length requirement, and they are represented by the black curves in the plots. These used a mother code exceeding the length requirement and punctured to match the nominal code word lengths while allowing for slightly higher deviations from the nominal code rate. At the end, additional simulation results for various punctured sets are included as to show demonstrate the ratecompatibilities of these codes. These sets are constructed via structured puncturing use an irregular structured mother code generated from the seed matrices and a puncture-degree sequence.

13 I C82.6e-4/264 Minimum Set of Simulations - Samsung length 52 code, ate.5 Nokia length 44 code, ate.5 Nokia length 52 code, ate b/no d Figure. Performance of n=52, r = /2 code. AWGN channel, PSK. - Samsung length 728 code, ate.667 Nokia length 78 code, ate.672 Nokia length 728 code, ate b/no d Figure 2. Performance of n=728, r = 2/3 code. AWGN channel, PSK. 2

14 I C82.6e-4/264 - Samsung length 234 code, ate.75 Nokia length 2296 code, ate.75 Nokia length 234 code, ate Additional Set of Simulations r = /2, n = 576, 728, b/no d Figure 3. Performance of n=234, r = 3/4 code. AWGN channel, PSK. - Samsung length 576 code, ate.5 Nokia length 572 code, ate.5 Nokia length 576 code, ate b/no d Figure 4. Performance of n=576, r = /2 code. AWGN channel, PSK. 3

15 I C82.6e-4/264 - Samsung length 728 code, ate.5 Nokia length 76 code, ate.5 Nokia length 728 code, ate b/no d Figure 5. Performance of n=728, r = /2 code. AWGN channel, PSK. - Samsung length 234 code, ate.5 Nokia length 2288 code, ate.5 Nokia length 234 code, ate b/no d r = 2/3, n = 576, 52, 234 Figure 6. Performance of n=234, r = /2 code. AWGN channel, PSK. 4

16 I C82.6e-4/264 - Samsung length 576 code, ate.667 Nokia length 6 code, ate.672 Nokia length 576 code, ate b/no d Figure 7. Performance of n=576, r = 2/3 code. AWGN channel, PSK. - Samsung length 52 code, ate.667 Nokia length 59 code, ate.672 Nokia length 52 code, ate b/no d Figure 8. Performance of n=52, r = 2/3 code. AWGN channel, PSK. 5

17 I C82.6e-4/264 - Samsung length 234 code, ate.667 Nokia length 238 code, ate.672 Nokia length 234 code, ate b/no d r = 3/4, n = 576, 52, 728 Figure 9. Performance of n=234, r = 2/3 code. AWGN channel, PSK. - Samsung length 576 code, ate.75 Nokia length 56 code, ate.75 Nokia length 576 code, ate b/no d Figure. Performance of n=576, r = 3/4 code. AWGN channel, PSK. 6

18 I C82.6e-4/264 - Samsung length 52 code, ate.75 Nokia length 2 code, ate.75 Nokia length 52 code, ate b/no d Figure. Performance of n=52, r = 3/4 code. AWGN channel, PSK. - Samsung length 728 code, ate.75 Nokia length 736 code, ate.75 Nokia length 728 code, ate b/no d Figure 2. Performance of n=728, r = 3/4 code. AWGN channel, PSK. 7

19 I C82.6e-4/264 ate-compatible Simulations: ate /2 Mother Code: sim punc output N234 M7 53Nokia45 5iters.mat ate.5, N=234, K=7 ate.527, N=222, K=7 ate.557, N=2, K=7 ate.59, N=98, K=7 ate.629, N=86, K=7 ate.672, N=742, K=7 ate.72, N=622, K=7 ate.779, N=52, K=7 ate.847, N=382, K=7 ate.927, N=262, K= b/no d Figure 3: ate /2 mother code using N SPAD = 45 and the following nested seed puncture-degree sequence. d = SD [,,,,,,,,,,,,5 4,, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 2, 3, 2, 3, 2, 2, 2, 3, 3, 2, 3, 2, 2, 2, 3, 2, 3, 2, 3, 2, 2, 3, 2, 3, 2, 3, 2, T 2, 2, 3, 3, 2, 2, 2, 3, 2, 2, 3, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3,,,2, 3, 2, 3, 2,,] 8

20 I C82.6e-4/264 ate 2/3 Mother Code xample: sim punc output N769 M58 6Nokia29 5iters.mat - -2 ate.672, N=769, K=89 ate.694, N=74, K=89 ate.77, N=658, K=89 ate.742, N=62, K=89 ate.769, N=546, K=89 ate.797, N=49, K=89 ate.829, N=435, K=89 ate.862, N=379, K=89 ate.899, N=323, K=89 ate.938, N=267, K= b/no d Figure 4: ate 2/3 mother code using N SPAD = 29 and the following nested seed puncture-degree sequence T d SD = [,, 3, 3, 3, 3, 3, 2, 3, 3, 2, 3, 3, 2, 2, 2, 2, ], 9

21 I C82.6e-4/264 ate 2/3 Mother Code xample: sim punc output N238 M76 6Nokia38 5iters.mat - -2 ate.672, N=238, K=558 ate.694, N=2245, K=558 ate.78, N=27, K=558 ate.743, N=297, K=558 ate.77, N=223, K=558 ate.799, N=95, K=558 ate.83, N=876, K=558 ate.865, N=82, K=558 ate.92, N=728, K=558 ate.942, N=654, K= b/no d Figure 5: ate 2/3 mother code using N SPAD = 38 and the following nested seed puncture-degree sequence T d SD = [,,, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 2, 3, 2, 3, 3, 3, 2, 3, 3, 2, 3, 2, 2, 2, 2, 2, 2,,, 3],2 2

22 I C82.6e-4/264 ate 3/4 Mother Code xample: sim punc output N2352 M588 56Nokia42 5iters.mat - -2 ate.75, N=2352, K=764 ate.768, N=2297, K=764 ate.787, N=2242, K=764 ate.87, N=286, K=764 ate.828, N=23, K=764 ate.85, N=276, K=764 ate.873, N=22, K=764 ate.898, N=965, K=764 ate.924, N=9, K=764 ate.95, N=854, K= b/no d Figure 6: ate 3/4 mother code using N SPAD = 42 and the following nested seed puncture-degree sequence. d SD = [,3 T,,, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 2, 3, 3, 2, 3, 3, 3, 2,, 2, 3, 2, 3, 2, 3, 2, 2, 2, 2, 2, 2,, 2] 2

23 I C82.6e-4/264 eferences: []. G. Gallager, ow-density parity-check codes, I Trans. Inform. Theory, vol. IT-8, pp.2-28, Jan [2] T. J. ichardson, M. A. Shokrollahi, and.. Urbanke, Design of Capacity Approaching Irregular ow-density Parity-Check Codes, I Transactions on Information Theory, vol. 47, pp , Feb. 2. [3] Sae-Young Chung, On the Construction of Some Capacity-Approaching Coding Schemes, PhD Dissertation, MIT, 2. [4] P. Joo, et al., DPC coding for OFDMA PHY, I C82.6d-4/86r, May 24. [5] T. J. ichardson and.. Urbanke, fficient ncoding of ow-density Parity-Check Codes, I Transactions on Information Theory, vol. 47, pp , Feb