Codes designed via algebraic lifts of graphs
|
|
- Emmeline Sparks
- 5 years ago
- Views:
Transcription
1 p./40 Codes designed via algebraic lifts of graphs Clemson Mini-Conference on Discrete Mathematics Oct. 3, Christine A. Kelley Department of Mathematics University of Nebraska-Lincoln
2 p.2/40 Outline Channel coding and LDPC codes Random vs. algebraic constructions Protograph codes Voltage graphs Guidelines for choosing voltage assignments Construction Voltage assignment scheme Examples of codes and their performance Extensions and conclusions
3 p.3/40 Channel Coding Goal: Communication in the presence of noise.
4 y p.3/40 x Channel Coding Goal: Communication in the presence of noise. x =(x,x,...,x ) 2 k Message x = (x,x 2,...,x k ) of length k. Components from alphabet F 2 = {0,}.
5 y p.3/40 x Channel Coding Goal: Communication in the presence of noise. x =(x,x 2,...,x k ) c =(c,c 2,...,c n) Encoder Encoder maps x to codeword c = (c,c 2,...,c n ) of length n, (n > k).
6 y p.3/40 x Channel Coding Goal: Communication in the presence of noise. x =(x,x,...,x ) c =(c,c,...,c n ) r =(r,r,...,r ) 2 k 2 2 Encoder Channel n Received word r = (r,r 2,...,r n ) of length n obtained by from the channel. r = c+error vector
7 y p.3/40 x Channel Coding Goal: Communication in the presence of noise. x =(x,x 2,...,x k ) c =(c,c 2,...,c n ) r =(r,r 2,...,r n) x =(x,x 2,...,x k ) Encoder Channel Decoder Decoder estimates codeword ĉ = (ĉ,ĉ 2,...,ĉ n ) to get estimate ˆx of x. ˆx? = x
8 y p.3/40 x Channel Coding Goal: Communication in the presence of noise. x =(x,x 2,...,x k ) c =(c,c 2,...,c n ) r =(r,r 2,...,r n) x =(x,x 2,...,x k ) Encoder Channel Decoder Code is image of all messages in F k 2 in the space Fn 2. Code is linear if code forms linear subpace of F n 2.
9 p.4/40 Linear Block Codes An [n,k,d] linear block code is a code having codewords of length n dimension k rate r = n k distance d = min c,c C {i c i c i } = min c C {i c i 0}.
10 p.4/40 Linear Block Codes An [n,k,d] linear block code is a code having codewords of length n dimension k rate r = n k distance d = min c,c C {i c i c i } = min c C {i c i 0}. Generator matrix (Encoder): G Mat k n (F 2 ) C = {c = xg x F k 2}.
11 p.4/40 Linear Block Codes An [n,k,d] linear block code is a code having codewords of length n dimension k rate r = n k distance d = min c,c C {i c i c i } = min c C {i c i 0}. Generator matrix (Encoder): G Mat k n (F 2 ) C = {c = xg x F k 2}. Parity-check matrix H is m n matrix such that GH T = 0. C = {c ch T = 0}.
12 p.5/40 Graph representation of a code Let C be a linear block code defined by the following parity-check matrix H. H = parity \bit x 0 x x 2 x 3 x 4 x 5 x 6 p p p p ch T = 0 (x 0,x,...,x 6 )H T = 0
13 p.6/40 Graph representation of a code We can represent H by the following bipartite graph. x 0 x x 2 p 0 x 3 x 4 x 5 p p 2 p 3 ch T = 0 (x 0,x,...,x 6 )H T = 0 x 6
14 p.6/40 Graph representation of a code We can represent H by the following bipartite graph. x 0 x x 2 p 0 p x 3 p 2 x 4 x 5 p3 p 0 : x 0 + x + x 2 + x 5 = 0. x 6 Code is the set of all binary vectors that, when input to the variable nodes, satisfy all the check equations
15 p.6/40 Graph representation of a code We can represent H by the following bipartite graph. x 0 x x 2 p 0 p x 3 p 2 x 4 x 5 p3 p : x 0 + x 2 + x 3 + x 4 = 0. x 6 Code is the set of all binary vectors that, when input to the variable nodes, satisfy all the check equations
16 p.6/40 Graph representation of a code We can represent H by the following bipartite graph. x 0 x x 2 p 0 p x 3 p 2 x 4 x 5 p3 p 2 : x + x 4 + x 5 + x 6 = 0. x 6 Code is the set of all binary vectors that, when input to the variable nodes, satisfy all the check equations
17 p.6/40 Graph representation of a code We can represent H by the following bipartite graph. x 0 x x 2 p 0 p x 3 p 2 x 4 x 5 p3 p 3 : x + x 3 + x 4 + x 6 = 0. x 6 Code is the set of all binary vectors that, when input to the variable nodes, satisfy all the check equations
18 p.7/40 Low Density Parity Check (LDPC) codes LDPC codes are a class of linear block codes characterized by having a sparse parity check matrix H. LDPC codes may be represented graphically via Tanner graphs. LDPC codes provide near-capacity performance while simultaneously admitting implementable decoders.
19 p.8/40 Graph properties that influence iterative decoder. Girth The girth is the smallest number of edges in the graph that form a cycle. 2. Diameter The diameter is the farthest distance between any pair of vertices 3. Expansion 4. Other: Stopping sets, trapping sets, etc.
20 p.9/40 Low Density Parity Check (LDPC) codes: History Gallager in 963 Margulis in the 970s Tanner in 98 MacKay and Neal in 995 Richardson, Urbanke, Shokrollahi in 996
21 p.0/40 Construction Techniques π Random. Regular random LDPCs have been shown to be asymptotically good codes their minimum distance d min grows linearly with block length. Irregular random LDPCs have been shown to perform very close to the theoretical limits at poor channel conditions. Degree profile of the LDPC constraint graph determines convergence behavior of the BP decoder.
22 p.0/40 Construction Techniques π Algebraic. Have structure. Efficient representation. Lower encoding complexity.
23 p./40 Protograph codes Protograph codes are codes obtained by taking lifts of a suitably chosen base graph and using the lift (or, derived graph) as the graph for the code. Idea: the structure of the base graph influences the structure of the graphs representing the codes.
24 p./40 Protograph codes Protograph codes are codes obtained by taking lifts of a suitably chosen base graph and using the lift (or, derived graph) as the graph for the code. Idea: the structure of the base graph influences the structure of the graphs representing the codes. Existing protograph constructions use random lifts, which can be interpreted as replacing s in the parity-check matrix by permutation matrices (array-based LDPC codes), and 0-matrices elsewhere. Other code families (e.g. Repeat-accumulate codes) also designed in this way.
25 Graph Lifts Sketch of graph with a degree 5 lift p.2/40
26 p.3/40 Constructing codes using algebraic lifts Idea: Start with a base graph and use algebraically obtained lifts of the base graph to represent codes. The lifts are determined by algebraic voltage assignments that are associated with the edges in the base graph. This voltage graph framework unifies several families of existing codes [with Walker]. Benefits Ability to prove properties of the resulting codes. Potential to analyze random protograph codes algebraically.
27 p.4/40 Voltage graphs [Gross and Tucker, 977] LetG = (V G,E G ) be a graph where each edge has an orientation, G a finite group, and α : E(T) G is a mapping from the positively oriented edges to G. Then G is an (ordinary) voltage graph and G α = (V α,e α ) is the derived graph, where V α = V G = {v g v V,g G}. E α = E G = {e g e E,g G}. If e + = (u,v) E and α(e + ) = a G, then e + g = (u g,v ga ) E α.
28 p.4/40 Voltage graphs [Gross and Tucker, 977] LetG = (V G,E G ) be a graph where each edge has an orientation, G a finite group, and α : E(T) G is a mapping from the positively oriented edges to G. Then G is an (ordinary) voltage graph and G α = (V α,e α ) is the derived graph, where V α = V G = {v g v V,g G}. E α = E G = {e g e E,g G}. If e + = (u,v) E and α(e + ) = a G, then e + g = (u g,v ga ) E α. If α(e + ) = a G, then α(e ) = a G. G is called the ordinary voltage group, and α is an ordinary voltage assignment.
29 p.5/40 Example: Ordinary voltage graph and derived graph The ordinary voltage group is Z/5Z. u 0 c 0 d 0 v 0 c 4 c u 0 v d e 2 u c d v e 4 e 0 e 2 e e 3 v 4 v 2 v 3 d 4 c 3 u 4 d 2 d 3 u 2 u 3 Remark: G α is a G -degree lift ofg, with natural projection map ρ :G α G. v g v, e g e c 2
30 p.6/40 Permutation voltage graph Let G be a permutation group, i.e. a subgroup of the symmetric group S n. LetG = (V G,E G ) be a graph where each edge has an orientation, and α : E(T) G is a mapping from the positively oriented edges to G. ThenG is a permutation voltage graph and G α = (V α,e α ) is the permutation derived graph where V α = V {,2,...,n} E α = E {,2,...,n}. If τ S n is a permutation voltage on the edge e + = (u,v) ing, then (u i,v τ(i) ) is an edge ing α.
31 p.6/40 Permutation voltage graph Let G be a permutation group, i.e. a subgroup of the symmetric group S n. LetG = (V G,E G ) be a graph where each edge has an orientation, and α : E(T) G is a mapping from the positively oriented edges to G. ThenG is a permutation voltage graph and G α = (V α,e α ) is the permutation derived graph where V α = V {,2,...,n} E α = E {,2,...,n}. If τ S n is a permutation voltage on the edge e + = (u,v) ing, then (u i,v τ(i) ) is an edge ing α. If α(e + ) = σ G, then α(e ) = σ G. Note thatg α is an n-degree lift ofg.
32 Example: Permutation voltage graph and derived graph The permutation voltage group is S 3. u 3 y 2 v3 z 3 x 3 x 2 u 2 y 3 v 2 x z 2 z u y v ( 2 3) x u ()(2 3) y v z ( 2)(3) Remark: G α is a degree 3 lift ofg, with natural projection map ρ :G α G. v i v, e i e, for i =,2,3 p.7/40
33 p.8/40 Walks in the base graph A walk ing is a sequence of oriented edges in the order they are traversed: W = e σ eσ 2 2 eσ n n, with σ i = {±} and e,...,e n E. The length of W is the number of edges in the walk. The net voltage of a walk W is the voltage group product of the voltages on the edges in W in the order and direction of W.
34 p.9/40 Net voltage example The walk W = z + y x + has net voltage ( 2)(3) ()(2 3) ( 2 3) = ()(3)(2) S 3 u 3 y 2 v3 z 3 x 3 x 2 u 2 y 3 v 2 x z 2 z u y v ( 2 3) x u ()(2 3) y v z ( 2)(3)
35 p.20/40 Example: SFT [55,64,20] code as a voltage graph- based code Set q = 3, k = 5, and j = 3. Choose a = 2 and b = 5 in F 3. o(2) = 5 and o(5) = 3 The parity-check matrix H is: H = I I 2 I 4 I 8 I 6 I 5 I 0 I 20 I 9 I 8 I 25 I 9 I 7 I 4 I 28 (93 55), where I x is a 3 3 identity matrix shifted to the left by x positions.
36 p.2/40 Example: Sridhara-Fuja-Tanner codes H = Example of a sparse parity check matrix representing an LDPC code: A 3 5 array of permutation matrices.
37 Example: Sridhara-Fuja-Tanner codes p.2/40
38 Example: Sridhara-Fuja-Tanner codes p.2/40
39 p.22/40 Previous work on choosing voltages Previous work: We classified subgraphs of base graphs that force cycles in derived graph under any abelian voltage assignments [K. and Walker, 2008]. IfG contains an (a,b,c)-theta graph or an (a,a 2 ;b)-dumbbell graph, then for any abelian voltage assignment α : E(G) G, a cycle will result in the derived graphg of length 2(a+b+c) for the theta-graph case. [see also, Exoo] 2(a + a 2 )+4b for the dumbbell-graph case. If a derived graph has an abelian-inevitable cycle (i.e. from abelian voltages), then the graphg contains either a theta graph or a dumbbell graph.
40 p.23/40 Choosing voltages (previous insights) Commuting permutations have been shown to limit girth: For j k array of commuting permutations, the code has girth 2 [Tanner, Sridhara, Fuja, 200.] Inevitable cycles result in derived graphs from (most) base graphs with any abelian voltage assignment to any abelian group. Commuting permutations have also been shown to limit distance: For j k array of commuting permutations, the code has distance ( j+ )! [MacKay and Davey, 999.] Guideline : Choose a nonabelian voltage group (and non-commuting voltages).
41 p.24/40 Cycle Decomposition of Permutations Every permutation has a unique cycle decomposition, for example σ = may be written as σ = (572)(34)(6) The cycle structure of a permutation π in S n is the vector (c,c 2,...,c n ) where c j is the number of j-cycles in the cycle decomposition of π. The cycle structure of σ is (,,0,,0,0,0)
42 p.25/40 Voltages to give good cycle structure Theorem (Gross and Tucker, 977) Let C be a cycle of length k in the base graph of a permutation voltage graph with net voltage π, and let (c,c 2,...,c n ) be the cycle structure of π. Then the pre-image of C in the derived graph has c + c 2 + +c n components, including for each j =,...,n, exactly c j cycles of length k j. The above result also extends to closed walks in G. Guideline 2: Choose voltages so that the net voltages of short cycles (and short closed walks) in G do not have small cycles in their cycle decompositions.
43 How to incorportate Guideline 2 in a design i v i i c v 2 c c 2 i v 3 d Choose voltages so that the net voltages of the 4-cycles ing and avoid cycles of length 3: c c cd dc d d Similarly, we check the closed walks of length 6 and avoid net voltages with cycles of length or 2, and for cycles or walks of length 8 or 2, we avoid fixed points in the net voltages. c 2 (c ) 2 d 2 (d ) 2 cd d c dc c d cdc c d c dcd d c d p.26/40
44 p.27/40 Voltages and connectivity Let α be a voltage assignment fromg to G. Then for any spanning treet ofg, there is an assignment α such that the derived graphg α = G and the edges oft are each assigned the identity element (Gross and Tucker). The edges outside of T form the co-tree, and their voltages generate the local voltage group, G.
45 p.27/40 Voltages and connectivity Let α be a voltage assignment fromg to G. Then for any spanning treet ofg, there is an assignment α such that the derived graphg α = G and the edges oft are each assigned the identity element (Gross and Tucker). The edges outside of T form the co-tree, and their voltages generate the local voltage group, G. The number of components of a permutation derived graphg α with such an assignment is equal to the number of orbits of the action of G on {,2,...,n}. Guideline 3: Choose a permutation group whose action yields one orbit, and assign identity element to the edges of a spanning tree and generating voltages to the edges of the co-tree.
46 p.28/40 Steps of the construction Our examples will use the complete bipartite graph K j,k as a base graph, though method may be applied to any base graph.. Choose base graph and orient edges from variables to checks, and choose a nonabelian group G of order m, and label the elements of G from to m.
47 p.28/40 Steps of the construction Our examples will use the complete bipartite graph K j,k as a base graph, though method may be applied to any base graph.. Choose base graph and orient edges from variables to checks, and choose a nonabelian group G of order m, and label the elements of G from to m. 2. Let G act on itself by left multiplication to obtain an isomorphic group P, where P is a permutation group of order m. The only element in P with a fixed point is the identity permutation.
48 p.28/40 Steps of the construction Our examples will use the complete bipartite graph K j,k as a base graph, though method may be applied to any base graph.. Choose base graph and orient edges from variables to checks, and choose a nonabelian group G of order m, and label the elements of G from to m. 2. Let G act on itself by left multiplication to obtain an isomorphic group P, where P is a permutation group of order m. The only element in P with a fixed point is the identity permutation. 3. Assign voltages Choose a spanning tree of the base graph and assign each edge in the tree the identity permutation i. For edges in co-tree, choose the nontrivial voltages so that each belongs to a distinct cyclic subgroup of the group P, and, in addition, the generators of the group should be chosen among this set.
49 p.29/40 Construction step Choose m = pq such that p and q are prime, q < p, and q (p ). We construct the nonabelian group G generated by elements c and d such that c = p and d = q and dc = c s d, where s (mod p) and s q (mod p).
50 p.29/40 Construction step Choose m = pq such that p and q are prime, q < p, and q (p ). We construct the nonabelian group G generated by elements c and d such that c = p and d = q and dc = c s d, where s (mod p) and s q (mod p). Example: Let p = 3 and q = 2 with relation c 2 d = dc. Then G = {,c,c 2,d,cd,c 2 d = dc}. Order the elements according to their position.,c 2,c 2 3,d 4,cd 5,c 2 d 6.
51 p.30/40 Step 2: Obtaining the permutation group Let each element of G act on G from the left to obtain a permutation group P of order m. Example: The action of c on G = {,c,c 2,d,cd,c 2 d = dc} yields the set {c g g G} = {c,c 2,,cd,c 2 d,d} = {2,3,,5,6,4}. This correponds to the permutation c = (23)(456)
52 p.30/40 Step 2: Obtaining the permutation group Let each element of G act on G from the left to obtain a permutation group P of order m. Example: The action of c on G = {,c,c 2,d,cd,c 2 d = dc} yields the set {c g g G} = {c,c 2,,cd,c 2 d,d} = {2,3,,5,6,4}. This correponds to the permutation c = (23)(456) Note: The action of P on {,2,...,n} has just one orbit by orbit-counting lemma: P σ P fix(σ).
53 p.3/40 Step 3: Assigning the voltages. Assign the identity permutation to the edges of a chosen spanning tree,t. We chooset to contain the eges corresponding to the first row and column of j k array. 2. Choose remaining ( j )(k ) nontrivial voltages so that Each is from a different cyclic subgroup of P. They generate P (for connectivity). Net voltages of short cycles are checked. Note that due to construction, the 8-cycles and 2-cycles will not have fixed points, unless the net voltage is the identity.
54 p.32/40 Code Construction with Order 55 To avoid 3-cycles or smaller in the net voltages, p,q > 3. Let G be the nonabelian group of order m = 55, with p = and q = 5. The relation is c 3 d = dc, where c = and d = 5. We generate the group, label the elements, and let G act on itself to form the isomorphic permutation group P with generators: c : (,2,3,4,5,6,7,8,9,0,) (2,6,7,8,9,20,2,22,23,24,25) (3,26,27,28,29,30,3,32,33,34,35) (4,36,37,38,39,40,4,42,43,44,45) (5,46,47,48,49,50,5,52,53,54,55). d : (,2,3,4,5)(2,8,34,40,49,2) (3,2,32,42,53)(4,24,30,39,46) (5,6,28,44,50)(6,9,26,38,54) (7,22,35,43,47)(8,25,33,37,5) (9,7,3,42,55)(0,20,36,48,0) (,23,27,4,52)
55 p.33/40 Code example I: (2,3) LDPC code Let i denote the identity permutation. H = i i i i c d The voltages are chosen from distinct cyclic subgroups of P and do not commute. The code has block length 65 and code rate The girth is 20. i v i i c v 2 c c 2 i v 3 d
56 p.34/40 Code example II: (3,5) LDPC code H = i i i i i i c c 2 d 2 c 5 d c 8 d 3 i d c 3 d 3 c 4 d 4 c 6 d 2 The code has block length 275 and code rate 0.4. The voltages are chosen from distinct cyclic subgroups of P and do not commute.
57 p.35/40 Simulation Results: (2,3) LDPC codes Performance of (2,3) LDPC codes over the memoryless additive white Gaussian noise (AWGN) channel New construction Array/SFT construction Random construction Randomly chosen permutations 50 sum product iterations Bit error rate (BER) SNR (db) Performance of block length 65, code rate 0.33, (2,3) LDPC codes on the binary-input additive white Gaussian noise channel under sum-product decoding.
58 p.36/40 Simulation Results: (3,5) LDPC codes Performance of (3,5) LDPC codes over the memoryless additive white Gaussian noise (AWGN) channel New construction Array/SFT construction Random construction Randomly chosen permutations 0 3 (50 sum product iterations) Bit error rate (BER) SNR (db) Performance of block length 275, code rate 0.4, (3,5) LDPC codes on the binary-input additive white Gaussian noise channel under sum-product decoding.
59 p.37/40 Extensions: code ensembles Next step: Use optimal base graph and generalize the voltage assignment to an arbitrary nonabelian group of order m = pq. Then prove asymptotic results about the ensemble.
60 p.38/40 Summary and Ongoing work We presented a construction that demonstrates that codes based on algebraic lifts have the potential to outperform random codes, as well as those based on random lifts. Find an explicit relationship between the voltages and distance. Use other nonabelian groups Wider range of permutation groups to make more block lengths possible. Groups with more generators will make the choices of voltages easier. Use ordinary voltage assignments. How voltages affect other parameters important for decoding: trapping sets, stopping sets. Extend to constructions of repeat-accumulate codes, interleavers, etc.
61 References [] J.L. Gross and T.W. Tucker, Topological graph theory, Wiley, New York, 987. [2] G. Exoo, Voltage graphs, group presentations, and cage", The Electronic Journal of Combinatorics, vol. (), [3] R. M. Tanner, D. Sridhara, and T. E. Fuja, A class of group-structured LDPC codes", in Proceedings of International Symposium on Communication Theory and Applications, Ambleside, U.K., pp , July 200. [4] L. Brankovic, M. Miller, J. Plesnik, J. Ryan, and J. Siran, Large graphs with small degree and diameter: A voltage assignment approach", Journal of Combinatorial Mathematics and Combinatorial Computing, vol. 24, pp. 6-76, 997 [5] R. M. Tanner, On quasi-cyclic repeat accumulate codes," in Proc. 37th Allerton Conf. Communication, Control, and Computing, Sept. 999, pp [6] J. Thorpe, LDPC codes constructed from protographs", IPN progress report, pp , JPL, [7] J. L. Fan, Array codes as low-density parity-check codes," in Proc. 2nd. Int. Symp. Turbo codes and related topics, pp , Brest, France, Sept. 4-7, [8] O. Milenkovic, N. Kashyap, D. Leyba, Shortened array codes of large girth", preprint on archive, [9] J. Thorpe, K. Andrews, and S. Dolinar, Methodologies for designing LDPC codes using protographs and circulants.", in Proceedings of the IEEE International Symposium on Information Theory, p. 236, Chicago, June [0] D. J. C. MacKay and M. C. Davey, Evaluation of Gallager codes for short block length and high rate applications", in Codes, Systems, and Graphical Models, B. Marcus and J. Rosenthal, Eds., vol. 23 of IMA Volumes in Mathematics and its Applications, pp Springer, New York, [] R. Smarandache and P.O. Vontobel, On regular quasi-cyclic LDPC codes from binomials", in Proceedings of the IEEE International Symposium on Information Theory, Chicago, IL, June [2] K. Yang and T. Helleseth, On the minimum distance of array codes as LDPC codes", in IEEE Transactions on Information Theory, vol. 49, No. 2, December p.39/40
62 p.40/40 Example: the Binary Erasure Channel (BEC) Input 0 p p Output 0 Binary input B E C Ternary output p p Note: There are no errors but only erasures or losses. The Internet is a real-world example for this channel.
63 p.40/40 Example: the Binary Erasure Channel (BEC) Received word = (0,,,,,,,) 0
64 p.40/40 Example: the Binary Erasure Channel (BEC) Received word = (0,,,,,,,) 0 0
65 p.40/40 Example: the Binary Erasure Channel (BEC) Received word = (0,,,,,,,) 0 0 0
66 p.40/40 Example: the Binary Erasure Channel (BEC) Received word = (0,,,,,,,) 0 0 0
67 p.40/40 Example: the Binary Erasure Channel (BEC) Received word = (0,,,,,,,)
68 p.40/40 Example: the Binary Erasure Channel (BEC) Received word = (0,,,,,,,) 0 0 0
69 p.40/40 Example: the Binary Erasure Channel (BEC) Received word = (0,,,,,,,) 0 0 0
70 p.40/40 Example: the Binary Erasure Channel (BEC) Received word = (0,,,,,,,)
71 p.40/40 Example: the Binary Erasure Channel (BEC) Received word = (0,,,,,,,) Estimate=(0, 0,,, 0,, 0, )
LDPC codes from voltage graphs
LDPC codes from voltage graphs Christine A. Kelley Department of Mathematics University of Nebraska-Lincoln Lincoln, NE 68588, USA. Email: ckelley2@math.unl.edu Judy L. Walker Department of Mathematics
More informationConstruction of low complexity Array based Quasi Cyclic Low density parity check (QC-LDPC) codes with low error floor
Construction of low complexity Array based Quasi Cyclic Low density parity check (QC-LDPC) codes with low error floor Pravin Salunkhe, Prof D.P Rathod Department of Electrical Engineering, Veermata Jijabai
More informationConstruction of Protographs for QC LDPC Codes With Girth Larger Than 12 1
Construction of Protographs for QC LDPC Codes With Girth Larger Than 12 1 Sunghwan Kim, Jong-Seon No School of Electrical Eng. & Com. Sci. Seoul National University, Seoul, Korea Email: {nodoubt, jsno}@snu.ac.kr
More informationPartially Quasi-Cyclic Protograph-Based LDPC Codes
Partially Quasi-Cyclic Protograph-Based LDPC Codes Roxana Smarandache Department of Mathematics and Statistics San Diego State University San Diego, CA 92182 Email: rsmarand@sciencessdsuedu David G M Mitchell
More informationIterative Encoding of Low-Density Parity-Check Codes
Iterative Encoding of Low-Density Parity-Check Codes David Haley, Alex Grant and John Buetefuer Institute for Telecommunications Research University of South Australia Mawson Lakes Blvd Mawson Lakes SA
More informationStructured Low-Density Parity-Check Codes: Algebraic Constructions
Structured Low-Density Parity-Check Codes: Algebraic Constructions Shu Lin Department of Electrical and Computer Engineering University of California, Davis Davis, California 95616 Email:shulin@ece.ucdavis.edu
More informationQuasi-Cyclic Asymptotically Regular LDPC Codes
2010 IEEE Information Theory Workshop - ITW 2010 Dublin Quasi-Cyclic Asymptotically Regular LDPC Codes David G. M. Mitchell, Roxana Smarandache, Michael Lentmaier, and Daniel J. Costello, Jr. Dept. of
More informationQuasi-Cyclic Low-Density Parity-Check Codes With Girth Larger Than
IEEE TRANSACTIONS ON INFORMATION THEORY, VOL 53, NO 8, AUGUST 2007 2885 n possible values If the parity check is satisfied, the error probability is closely approximated by the probability of two bit errors,
More informationDesign of Non-Binary Quasi-Cyclic LDPC Codes by Absorbing Set Removal
Design of Non-Binary Quasi-Cyclic LDPC Codes by Absorbing Set Removal Behzad Amiri Electrical Eng. Department University of California, Los Angeles Los Angeles, USA Email: amiri@ucla.edu Jorge Arturo Flores
More informationRECURSIVE CONSTRUCTION OF (J, L) QC LDPC CODES WITH GIRTH 6. Communicated by Dianhua Wu. 1. Introduction
Transactions on Combinatorics ISSN (print: 2251-8657, ISSN (on-line: 2251-8665 Vol 5 No 2 (2016, pp 11-22 c 2016 University of Isfahan wwwcombinatoricsir wwwuiacir RECURSIVE CONSTRUCTION OF (J, L QC LDPC
More informationConstructions of Nonbinary Quasi-Cyclic LDPC Codes: A Finite Field Approach
Constructions of Nonbinary Quasi-Cyclic LDPC Codes: A Finite Field Approach Shu Lin, Shumei Song, Lan Lan, Lingqi Zeng and Ying Y Tai Department of Electrical & Computer Engineering University of California,
More informationIntroduction to Low-Density Parity Check Codes. Brian Kurkoski
Introduction to Low-Density Parity Check Codes Brian Kurkoski kurkoski@ice.uec.ac.jp Outline: Low Density Parity Check Codes Review block codes History Low Density Parity Check Codes Gallager s LDPC code
More informationCodes on Graphs. Telecommunications Laboratory. Alex Balatsoukas-Stimming. Technical University of Crete. November 27th, 2008
Codes on Graphs Telecommunications Laboratory Alex Balatsoukas-Stimming Technical University of Crete November 27th, 2008 Telecommunications Laboratory (TUC) Codes on Graphs November 27th, 2008 1 / 31
More informationAn Introduction to Low Density Parity Check (LDPC) Codes
An Introduction to Low Density Parity Check (LDPC) Codes Jian Sun jian@csee.wvu.edu Wireless Communication Research Laboratory Lane Dept. of Comp. Sci. and Elec. Engr. West Virginia University June 3,
More informationStopping, and Trapping Set Analysis
LDPC Codes Based on Latin Squares: Cycle Structure, Stopping, and Trapping Set Analysis Stefan Laendner and Olgica Milenkovic Electrical and Computer Engineering Department University of Colorado, Boulder,
More informationQuasi-cyclic Low Density Parity Check codes with high girth
Quasi-cyclic Low Density Parity Check codes with high girth, a work with Marta Rossi, Richard Bresnan, Massimilliano Sala Summer Doctoral School 2009 Groebner bases, Geometric codes and Order Domains Dept
More informationAn Efficient Algorithm for Finding Dominant Trapping Sets of LDPC Codes
An Efficient Algorithm for Finding Dominant Trapping Sets of LDPC Codes Mehdi Karimi, Student Member, IEEE and Amir H. Banihashemi, Senior Member, IEEE Abstract arxiv:1108.4478v2 [cs.it] 13 Apr 2012 This
More informationA Class of Quantum LDPC Codes Derived from Latin Squares and Combinatorial Design
A Class of Quantum LDPC Codes Derived from Latin Squares and Combinatorial Design Salah A Aly Department of Computer Science, Texas A&M University, College Station, TX 77843-3112, USA Email: salah@cstamuedu
More informationLDPC Codes. Slides originally from I. Land p.1
Slides originally from I. Land p.1 LDPC Codes Definition of LDPC Codes Factor Graphs to use in decoding Decoding for binary erasure channels EXIT charts Soft-Output Decoding Turbo principle applied to
More informationPerformance Analysis and Code Optimization of Low Density Parity-Check Codes on Rayleigh Fading Channels
Performance Analysis and Code Optimization of Low Density Parity-Check Codes on Rayleigh Fading Channels Jilei Hou, Paul H. Siegel and Laurence B. Milstein Department of Electrical and Computer Engineering
More informationOn the Exhaustion and Elimination of Trapping Sets: Algorithms & The Suppressing Effect
On the Exhaustion and Elimination of Trapping Sets: Algorithms & The Suppressing Effect Chih-Chun Wang Center for Wireless Systems and Applications (CWSA) School of ECE, Purdue University, West Lafayette,
More informationPseudocodewords from Bethe Permanents
Pseudocodewords from Bethe Permanents Roxana Smarandache Departments of Mathematics and Electrical Engineering University of Notre Dame Notre Dame, IN 46556 USA rsmarand@ndedu Abstract It was recently
More informationAbsorbing Set Spectrum Approach for Practical Code Design
Absorbing Set Spectrum Approach for Practical Code Design Jiadong Wang, Lara Dolecek, Zhengya Zhang and Richard Wesel wjd@ee.ucla.edu, dolecek@ee.ucla.edu, zhengya@eecs.umich.edu, wesel@ee.ucla.edu Abstract
More informationLDPC Codes. Intracom Telecom, Peania
LDPC Codes Alexios Balatsoukas-Stimming and Athanasios P. Liavas Technical University of Crete Dept. of Electronic and Computer Engineering Telecommunications Laboratory December 16, 2011 Intracom Telecom,
More informationConstruction of Type-II QC LDPC Codes Based on Perfect Cyclic Difference Set
Chinese Journal of Electronics Vol24, No1, Jan 2015 Construction of Type-II QC LDPC Codes Based on Perfect Cyclic Difference Set ZHANG Lijun 1,LIBing 2 and CHENG Leelung 3 (1 School of Electronic and Information
More informationGraph-based codes for flash memory
1/28 Graph-based codes for flash memory Discrete Mathematics Seminar September 3, 2013 Katie Haymaker Joint work with Professor Christine Kelley University of Nebraska-Lincoln 2/28 Outline 1 Background
More informationGirth Analysis of Polynomial-Based Time-Invariant LDPC Convolutional Codes
IWSSIP 212, 11-13 April 212, Vienna, Austria ISBN 978-3-2-2328-4 Girth Analysis of Polynomial-Based Time-Invariant LDPC Convolutional Codes Hua Zhou and Norbert Goertz Institute of Telecommunications Vienna
More informationECEN 655: Advanced Channel Coding
ECEN 655: Advanced Channel Coding Course Introduction Henry D. Pfister Department of Electrical and Computer Engineering Texas A&M University ECEN 655: Advanced Channel Coding 1 / 19 Outline 1 History
More informationWeaknesses of Margulis and Ramanujan Margulis Low-Density Parity-Check Codes
Electronic Notes in Theoretical Computer Science 74 (2003) URL: http://www.elsevier.nl/locate/entcs/volume74.html 8 pages Weaknesses of Margulis and Ramanujan Margulis Low-Density Parity-Check Codes David
More informationCodes on graphs and iterative decoding
Codes on graphs and iterative decoding Bane Vasić Error Correction Coding Laboratory University of Arizona Prelude Information transmission 0 0 0 0 0 0 Channel Information transmission signal 0 0 threshold
More informationAchieving Flexibility in LDPC Code Design by Absorbing Set Elimination
Achieving Flexibility in LDPC Code Design by Absorbing Set Elimination Jiajun Zhang, Jiadong Wang, Shayan Garani Srinivasa, Lara Dolecek Department of Electrical Engineering, University of California,
More informationFrom Stopping sets to Trapping sets
From Stopping sets to Trapping sets The Exhaustive Search Algorithm & The Suppressing Effect Chih-Chun Wang School of Electrical & Computer Engineering Purdue University Wang p. 1/21 Content Good exhaustive
More informationON THE MINIMUM DISTANCE OF NON-BINARY LDPC CODES. Advisor: Iryna Andriyanova Professor: R.. udiger Urbanke
ON THE MINIMUM DISTANCE OF NON-BINARY LDPC CODES RETHNAKARAN PULIKKOONATTU ABSTRACT. Minimum distance is an important parameter of a linear error correcting code. For improved performance of binary Low
More informationLower Bounds on the Graphical Complexity of Finite-Length LDPC Codes
Lower Bounds on the Graphical Complexity of Finite-Length LDPC Codes Igal Sason Department of Electrical Engineering Technion - Israel Institute of Technology Haifa 32000, Israel 2009 IEEE International
More informationAdaptive Cut Generation for Improved Linear Programming Decoding of Binary Linear Codes
Adaptive Cut Generation for Improved Linear Programming Decoding of Binary Linear Codes Xiaojie Zhang and Paul H. Siegel University of California, San Diego, La Jolla, CA 9093, U Email:{ericzhang, psiegel}@ucsd.edu
More informationSpatially Coupled LDPC Codes
Spatially Coupled LDPC Codes Kenta Kasai Tokyo Institute of Technology 30 Aug, 2013 We already have very good codes. Efficiently-decodable asymptotically capacity-approaching codes Irregular LDPC Codes
More informationDesign of regular (2,dc)-LDPC codes over GF(q) using their binary images
Design of regular (2,dc)-LDPC codes over GF(q) using their binary images Charly Poulliat, Marc Fossorier, David Declercq To cite this version: Charly Poulliat, Marc Fossorier, David Declercq. Design of
More informationIEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 64, NO. 10, OCTOBER
IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 64, NO. 10, OCTOBER 2016 4029 Optimized Design of Finite-Length Separable Circulant-Based Spatially-Coupled Codes: An Absorbing Set-Based Analysis Behzad Amiri,
More informationTrapping Set Enumerators for Specific LDPC Codes
Trapping Set Enumerators for Specific LDPC Codes Shadi Abu-Surra Samsung Telecommunications America 1301 E. Lookout Dr. Richardson TX 75082 Email: sasurra@sta.samsung.com David DeClercq ETIS ENSEA/UCP/CNRS
More informationOn the Girth of (3,L) Quasi-Cyclic LDPC Codes based on Complete Protographs
On the Girth o (3,L) Quasi-Cyclic LDPC Codes based on Complete Protographs Sudarsan V S Ranganathan, Dariush Divsalar and Richard D Wesel Department o Electrical Engineering, University o Caliornia, Los
More informationIntegrated Code Design for a Joint Source and Channel LDPC Coding Scheme
Integrated Code Design for a Joint Source and Channel LDPC Coding Scheme Hsien-Ping Lin Shu Lin and Khaled Abdel-Ghaffar Department of Electrical and Computer Engineering University of California Davis
More informationEfficient design of LDPC code using circulant matrix and eira code Seul-Ki Bae
Efficient design of LDPC code using circulant matrix and eira code Seul-Ki Bae The Graduate School Yonsei University Department of Electrical and Electronic Engineering Efficient design of LDPC code using
More informationNon-binary Hybrid LDPC Codes: structure, decoding and optimization
Non-binary Hybrid LDPC Codes: structure, decoding and optimization Lucile Sassatelli and David Declercq ETIS - ENSEA/UCP/CNRS UMR-8051 95014 Cergy-Pontoise, France {sassatelli, declercq}@ensea.fr Abstract
More informationRecent Results on Capacity-Achieving Codes for the Erasure Channel with Bounded Complexity
26 IEEE 24th Convention of Electrical and Electronics Engineers in Israel Recent Results on Capacity-Achieving Codes for the Erasure Channel with Bounded Complexity Igal Sason Technion Israel Institute
More informationSTUDY OF PERMUTATION MATRICES BASED LDPC CODE CONSTRUCTION
EE229B PROJECT REPORT STUDY OF PERMUTATION MATRICES BASED LDPC CODE CONSTRUCTION Zhengya Zhang SID: 16827455 zyzhang@eecs.berkeley.edu 1 MOTIVATION Permutation matrices refer to the square matrices with
More informationPseudocodewords of Tanner Graphs
SUBMITTED TO IEEE TRANSACTIONS ON INFORMATION THEORY 1 Pseudocodewords of Tanner Graphs arxiv:cs/0504013v4 [cs.it] 18 Aug 2007 Christine A. Kelley Deepak Sridhara Department of Mathematics Seagate Technology
More informationLow-Density Parity-Check codes An introduction
Low-Density Parity-Check codes An introduction c Tilo Strutz, 2010-2014,2016 June 9, 2016 Abstract Low-density parity-check codes (LDPC codes) are efficient channel coding codes that allow transmission
More informationConvergence analysis for a class of LDPC convolutional codes on the erasure channel
Convergence analysis for a class of LDPC convolutional codes on the erasure channel Sridharan, Arvind; Lentmaier, Michael; Costello Jr., Daniel J.; Zigangirov, Kamil Published in: [Host publication title
More informationOn the minimum distance of LDPC codes based on repetition codes and permutation matrices
On the minimum distance of LDPC codes based on repetition codes and permutation matrices Fedor Ivanov Email: fii@iitp.ru Institute for Information Transmission Problems, Russian Academy of Science XV International
More informationResearch Article Structured LDPC Codes over Integer Residue Rings
Hindawi Publishing Corporation EURASIP Journal on Wireless Communications and Networking Volume 2008, Article ID 598401, 9 pages doi:10.1155/2008/598401 Research Article Structured LDPC Codes over Integer
More informationMaximum Likelihood Decoding of Codes on the Asymmetric Z-channel
Maximum Likelihood Decoding of Codes on the Asymmetric Z-channel Pål Ellingsen paale@ii.uib.no Susanna Spinsante s.spinsante@univpm.it Angela Barbero angbar@wmatem.eis.uva.es May 31, 2005 Øyvind Ytrehus
More informationOn Generalized EXIT Charts of LDPC Code Ensembles over Binary-Input Output-Symmetric Memoryless Channels
2012 IEEE International Symposium on Information Theory Proceedings On Generalied EXIT Charts of LDPC Code Ensembles over Binary-Input Output-Symmetric Memoryless Channels H Mamani 1, H Saeedi 1, A Eslami
More informationSome Applications of pq-groups in Graph Theory
Some Applications of pq-groups in Graph Theory Geoffrey Exoo Department of Mathematics and Computer Science Indiana State University Terre Haute, IN 47809 g-exoo@indstate.edu January 25, 2002 Abstract
More informationA Class of Quantum LDPC Codes Constructed From Finite Geometries
A Class of Quantum LDPC Codes Constructed From Finite Geometries Salah A Aly Department of Computer Science, Texas A&M University College Station, TX 77843, USA Email: salah@cstamuedu arxiv:07124115v3
More informationDesign and Analysis of Graph-based Codes Using Algebraic Lifts and Decoding Networks
University of Nebraska - Lincoln DigitalCommons@University of Nebraska - Lincoln Dissertations, Theses, and Student Research Papers in Mathematics Mathematics, Department of 3-2018 Design and Analysis
More informationOn the minimum distance of LDPC codes based on repetition codes and permutation matrices 1
Fifteenth International Workshop on Algebraic and Combinatorial Coding Theory June 18-24, 216, Albena, Bulgaria pp. 168 173 On the minimum distance of LDPC codes based on repetition codes and permutation
More informationOn the Typicality of the Linear Code Among the LDPC Coset Code Ensemble
5 Conference on Information Sciences and Systems The Johns Hopkins University March 16 18 5 On the Typicality of the Linear Code Among the LDPC Coset Code Ensemble C.-C. Wang S.R. Kulkarni and H.V. Poor
More informationTime-invariant LDPC convolutional codes
Time-invariant LDPC convolutional codes Dimitris Achlioptas, Hamed Hassani, Wei Liu, and Rüdiger Urbanke Department of Computer Science, UC Santa Cruz, USA Email: achlioptas@csucscedu Department of Computer
More informationAn algorithm to improve the error rate performance of Accumulate-Repeat-Accumulate codes Tae-Ui Kim
An algorithm to improve the error rate performance of Accumulate-Repeat-Accumulate codes Tae-Ui Kim The Graduate School Yonsei University Department of Electrical and Electronic Engineering An algorithm
More informationOn Bit Error Rate Performance of Polar Codes in Finite Regime
On Bit Error Rate Performance of Polar Codes in Finite Regime A. Eslami and H. Pishro-Nik Abstract Polar codes have been recently proposed as the first low complexity class of codes that can provably achieve
More informationLecture 4 : Introduction to Low-density Parity-check Codes
Lecture 4 : Introduction to Low-density Parity-check Codes LDPC codes are a class of linear block codes with implementable decoders, which provide near-capacity performance. History: 1. LDPC codes were
More informationLDPC codes based on Steiner quadruple systems and permutation matrices
Fourteenth International Workshop on Algebraic and Combinatorial Coding Theory September 7 13, 2014, Svetlogorsk (Kaliningrad region), Russia pp. 175 180 LDPC codes based on Steiner quadruple systems and
More informationGlobally Coupled LDPC Codes
Globally Coupled LDPC Codes Juane Li 1, Shu Lin 1, Khaled Abdel-Ghaffar 1, William E Ryan 2, and Daniel J Costello, Jr 3 1 University of California, Davis, CA 95616 2 Zeta Associates, Fairfax, VA 22030
More informationOn the Block Error Probability of LP Decoding of LDPC Codes
On the Block Error Probability of LP Decoding of LDPC Codes Ralf Koetter CSL and Dept. of ECE University of Illinois at Urbana-Champaign Urbana, IL 680, USA koetter@uiuc.edu Pascal O. Vontobel Dept. of
More informationDecoding of LDPC codes with binary vector messages and scalable complexity
Downloaded from vbn.aau.dk on: marts 7, 019 Aalborg Universitet Decoding of LDPC codes with binary vector messages and scalable complexity Lechner, Gottfried; Land, Ingmar; Rasmussen, Lars Published in:
More informationA Universal Theory of Pseudocodewords
A Universal Theory of Pseudocodewords Nathan Axvig, Emily Price, Eric Psota, Deanna Turk, Lance C. Pérez, and Judy L. Walker Abstract Three types of pseudocodewords for LDPC codes are found in the literature:
More informationAnalysis of Absorbing Sets and Fully Absorbing Sets of Array-Based LDPC Codes
Analysis of Absorbing Sets and Fully Absorbing Sets of Array-Based LDPC Codes Lara Dolecek, Zhengya Zhang, Venkat Anantharam, Martin J. Wainwright, and Borivoje Nikolić dolecek@mit.edu; {zyzhang,ananth,wainwrig,bora}@eecs.berkeley.edu
More informationError Floors of LDPC Coded BICM
Electrical and Computer Engineering Conference Papers, Posters and Presentations Electrical and Computer Engineering 2007 Error Floors of LDPC Coded BICM Aditya Ramamoorthy Iowa State University, adityar@iastate.edu
More informationAnalysis of Sum-Product Decoding of Low-Density Parity-Check Codes Using a Gaussian Approximation
IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 47, NO. 2, FEBRUARY 2001 657 Analysis of Sum-Product Decoding of Low-Density Parity-Check Codes Using a Gaussian Approximation Sae-Young Chung, Member, IEEE,
More informationOptimal Rate and Maximum Erasure Probability LDPC Codes in Binary Erasure Channel
Optimal Rate and Maximum Erasure Probability LDPC Codes in Binary Erasure Channel H. Tavakoli Electrical Engineering Department K.N. Toosi University of Technology, Tehran, Iran tavakoli@ee.kntu.ac.ir
More informationLow-density parity-check codes
Low-density parity-check codes From principles to practice Dr. Steve Weller steven.weller@newcastle.edu.au School of Electrical Engineering and Computer Science The University of Newcastle, Callaghan,
More informationCHAPTER 3 LOW DENSITY PARITY CHECK CODES
62 CHAPTER 3 LOW DENSITY PARITY CHECK CODES 3. INTRODUCTION LDPC codes were first presented by Gallager in 962 [] and in 996, MacKay and Neal re-discovered LDPC codes.they proved that these codes approach
More informationIntroducing Low-Density Parity-Check Codes
Introducing Low-Density Parity-Check Codes Sarah J. Johnson School of Electrical Engineering and Computer Science The University of Newcastle Australia email: sarah.johnson@newcastle.edu.au Topic 1: Low-Density
More informationSPA decoding on the Tanner graph
SPA decoding on the Tanner graph x,(i) q j,l = P(v l = x check sums A l \ {h j } at the ith iteration} x,(i) σ j,l = Σ P(s = 0 v = x,{v : t B(h )\{l}}) q {vt : t B(h j )\{l}} j l t j t B(h j )\{l} j,t
More informationLow-density parity-check (LDPC) codes
Low-density parity-check (LDPC) codes Performance similar to turbo codes Do not require long interleaver to achieve good performance Better block error performance Error floor occurs at lower BER Decoding
More informationPerformance Comparison of LDPC Codes Generated With Various Code-Construction Methods
Performance Comparison of LDPC Codes Generated With Various Code-Construction Methods Zsolt Polgar, Florin rdelean, Mihaly Varga, Vasile Bota bstract Finding good LDPC codes for high speed mobile transmissions
More informationResearch Letter Design of Short, High-Rate DVB-S2-Like Semi-Regular LDPC Codes
Research Letters in Communications Volume 2008, Article ID 324503, 4 pages doi:0.55/2008/324503 Research Letter Design of Short, High-Rate DVB-S2-Like Semi-Regular LDPC Codes Luca Barletta and Arnaldo
More informationTHIS paper provides a general technique for constructing
Protograph-Based Raptor-Like LDPC Codes for the Binary Erasure Channel Kasra Vakilinia Department of Electrical Engineering University of California, Los Angeles Los Angeles, California 90024 Email: vakiliniak@ucla.edu
More informationIEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 58, NO. 2, FEBRUARY
IEEE TRANSACTIONS ON INFORMATION THEORY, VOL 58, NO 2, FEBRUARY 2012 585 Quasi-Cyclic LDPC Codes: Influence of Proto- and Tanner-Graph Structure on Minimum Hamming Distance Upper Bounds Roxana Smarandache,
More informationBounds on Achievable Rates of LDPC Codes Used Over the Binary Erasure Channel
Bounds on Achievable Rates of LDPC Codes Used Over the Binary Erasure Channel Ohad Barak, David Burshtein and Meir Feder School of Electrical Engineering Tel-Aviv University Tel-Aviv 69978, Israel Abstract
More informationGALLAGER S binary low-density parity-check (LDPC)
1560 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 57, NO. 6, JUNE 2009 Group-Theoretic Analysis of Cayley-Graph-Based Cycle GF(2 p )Codes Jie Huang, Shengli Zhou, Member, IEEE, Jinkang Zhu, Senior Member,
More informationTwo-Bit Message Passing Decoders for LDPC. Codes Over the Binary Symmetric Channel
Two-Bit Message Passing Decoders for LDPC 1 Codes Over the Binary Symmetric Channel Lucile Sassatelli, Member, IEEE, Shashi Kiran Chilappagari, Member, IEEE, Bane Vasic, arxiv:0901.2090v3 [cs.it] 7 Mar
More informationAalborg Universitet. Bounds on information combining for parity-check equations Land, Ingmar Rüdiger; Hoeher, A.; Huber, Johannes
Aalborg Universitet Bounds on information combining for parity-check equations Land, Ingmar Rüdiger; Hoeher, A.; Huber, Johannes Published in: 2004 International Seminar on Communications DOI link to publication
More informationCapacity-Achieving Ensembles for the Binary Erasure Channel With Bounded Complexity
Capacity-Achieving Ensembles for the Binary Erasure Channel With Bounded Complexity Henry D. Pfister, Member, Igal Sason, Member, and Rüdiger Urbanke Abstract We present two sequences of ensembles of non-systematic
More informationModern Coding Theory. Daniel J. Costello, Jr School of Information Theory Northwestern University August 10, 2009
Modern Coding Theory Daniel J. Costello, Jr. Coding Research Group Department of Electrical Engineering University of Notre Dame Notre Dame, IN 46556 2009 School of Information Theory Northwestern University
More informationEnhancing Binary Images of Non-Binary LDPC Codes
Enhancing Binary Images of Non-Binary LDPC Codes Aman Bhatia, Aravind R Iyengar, and Paul H Siegel University of California, San Diego, La Jolla, CA 92093 0401, USA Email: {a1bhatia, aravind, psiegel}@ucsdedu
More informationCoding Techniques for Data Storage Systems
Coding Techniques for Data Storage Systems Thomas Mittelholzer IBM Zurich Research Laboratory /8 Göttingen Agenda. Channel Coding and Practical Coding Constraints. Linear Codes 3. Weight Enumerators and
More informationCirculant Arrays on Cyclic Subgroups of Finite Fields: Rank Analysis and Construction of Quasi-Cyclic LDPC Codes
Circulant Arrays on Cyclic Subgroups of Finite Fields: Rank Analysis and Construction of Quasi-Cyclic LDPC Codes 1 arxiv:10041184v1 [csit] 7 Apr 2010 Li Zhang 1, Shu Lin 1, Khaled Abdel-Ghaffar 1, Zhi
More informationConstruction of LDPC codes
Construction of LDPC codes Telecommunications Laboratory Alex Balatsoukas-Stimming Technical University of Crete July 1, 2009 Telecommunications Laboratory (TUC) Construction of LDPC codes July 1, 2009
More informationMessage-Passing Decoding for Low-Density Parity-Check Codes Harish Jethanandani and R. Aravind, IIT Madras
Message-Passing Decoding for Low-Density Parity-Check Codes Harish Jethanandani and R. Aravind, IIT Madras e-mail: hari_jethanandani@yahoo.com Abstract Low-density parity-check (LDPC) codes are discussed
More informationMessage Passing Algorithm with MAP Decoding on Zigzag Cycles for Non-binary LDPC Codes
Message Passing Algorithm with MAP Decoding on Zigzag Cycles for Non-binary LDPC Codes Takayuki Nozaki 1, Kenta Kasai 2, Kohichi Sakaniwa 2 1 Kanagawa University 2 Tokyo Institute of Technology July 12th,
More informationRandom Redundant Soft-In Soft-Out Decoding of Linear Block Codes
Random Redundant Soft-In Soft-Out Decoding of Linear Block Codes Thomas R. Halford and Keith M. Chugg Communication Sciences Institute University of Southern California Los Angeles, CA 90089-2565 Abstract
More informationIEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 56, NO. 1, JANUARY
IEEE TRANSACTIONS ON INFORMATION THEORY, VOL 56, NO 1, JANUARY 2010 181 Analysis of Absorbing Sets Fully Absorbing Sets of Array-Based LDPC Codes Lara Dolecek, Member, IEEE, Zhengya Zhang, Member, IEEE,
More informationTurbo Codes are Low Density Parity Check Codes
Turbo Codes are Low Density Parity Check Codes David J. C. MacKay July 5, 00 Draft 0., not for distribution! (First draft written July 5, 998) Abstract Turbo codes and Gallager codes (also known as low
More informationSpatially Coupled LDPC Codes Constructed from Protographs
IEEE TRANSACTIONS ON INFORMATION THEORY (SUBMITTED PAPER) 1 Spatially Coupled LDPC Codes Constructed from Protographs David G. M. Mitchell, Member, IEEE, Michael Lentmaier, Senior Member, IEEE, and Daniel
More informationEE229B - Final Project. Capacity-Approaching Low-Density Parity-Check Codes
EE229B - Final Project Capacity-Approaching Low-Density Parity-Check Codes Pierre Garrigues EECS department, UC Berkeley garrigue@eecs.berkeley.edu May 13, 2005 Abstract The class of low-density parity-check
More informationThe New Multi-Edge Metric-Constrained PEG/QC-PEG Algorithms for Designing the Binary LDPC Codes With Better Cycle-Structures
HE et al.: THE MM-PEGA/MM-QC-PEGA DESIGN THE LDPC CODES WITH BETTER CYCLE-STRUCTURES 1 arxiv:1605.05123v1 [cs.it] 17 May 2016 The New Multi-Edge Metric-Constrained PEG/QC-PEG Algorithms for Designing the
More informationFrom Product Codes to Structured Generalized LDPC Codes
From Product Codes to Structured Generalized LDPC Codes Michael Lentmaier, Gianluigi Liva, Enrico Paolini, and Gerhard Fettweis Vodafone Chair Mobile Communications Systems, Dresden University of Technology
More informationLow-Density Arrays of Circulant Matrices: Rank and Row-Redundancy Analysis, and Quasi-Cyclic LDPC Codes
Low-Density Arrays of Circulant Matrices: 1 Rank and Row-Redundancy Analysis, and Quasi-Cyclic LDPC Codes Qin Huang 1 and Keke Liu 2 and Zulin Wang 1 arxiv:12020702v1 [csit] 3 Feb 2012 1 School of Electronic
More informationAn Introduction to Low-Density Parity-Check Codes
An Introduction to Low-Density Parity-Check Codes Paul H. Siegel Electrical and Computer Engineering University of California, San Diego 5/ 3/ 7 Copyright 27 by Paul H. Siegel Outline Shannon s Channel
More information