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ON THE ANALYSIS AND APPLICATION OF LDPC CODES . OLGICA MILENKOVIC UNIVERSITY OF COLORADO, BOULDER A joint work with: VIDYA KUMAR (Ph.D) STEFAN LAENDNER (Ph.D) DAVID LEYBA (Ph.D) VIJAY NAGARAJAN (MS)

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ON THE ANALYSIS AND APPLICATION OF LDPC CODES OLGICA MILENKOVIC UNIVERSITY OF COLORADO, BOULDER A joint work with: VIDYA KUMAR (Ph.D) STEFAN LAENDNER (Ph.D) DAVID LEYBA (Ph.D) VIJAY NAGARAJAN (MS) KIRAN PRAKASH (MS)

OUTLINE A brief prologue to codes on charts A diagram of known results: arbitrary like codes for standard channel models Code structures amiable for down to earth execution: organized LDPC Codes agreeable for usage with great blunder floor properties Code plan for non-standard Channels with memory Asymmetric channels Applying the turbo-unraveling rule to established logarithmic codes: Reed-Solomon, Reed-Muller, BCH… Applying the turbo-translating rule to frameworks with unequal mistake insurance prerequisites

THE ERROR-CONTROL PARADIGM Noisy channels offer ascent to information blunders : transmission or capacity frameworks Need intense mistake control coding (ECC) plans : straight or non-direct Linear EC Codes : Generated through basic generator or equality check grid Binary data vector (length k ) Code vector (word): (length n ) Key property: "Least separation of the code", , littlest partition between two codewords Rate of the code R = k/n Binary direct codes:

LDPC CODES More than 40 years of research (1948-1994) revolved around Weights of blunders that a code is ensured to revise "Limited separation interpreting" can\'t accomplish Shannon constrain Trade-off least separation for productive deciphering Low-Density Parity-Check (LDPC) Codes Gallager 1963, Tanner 1984, MacKay 1996 1. Straight square codes with scanty (little portion of ones) equality check network 2. Have normal representation as far as bipartite charts 3. Basic and proficient iterative unraveling as conviction engendering (Pearl, 1980-1990)

THE CODE GRAPH AND ITERATIVE DECODING Most essential result of graphical portrayal: effective iterative disentangling Message passing: Variable hubs: convey to check hubs their unwavering quality (log-probabilities) Check hubs: choose which factors are not solid and "stifle" their data sources Small number of edges in chart = low multifaceted nature Nodes on left/right with consistent degree: normal code Otherwise, codes named sporadic Can conform "degree circulation" of factors/checks Variable hubs Check hubs (Irregular degrees/codes) Best execution over standard channels: long, unpredictable, arbitrary like LDPC codes Have corresponding to length of code, however amend numerous more mistakes

DECODING OF LDPC CODES Iterative translating ideal just if the code diagram has no cycles Vardy et.al. 1997: All "great codes" must have cycles What are alluring code properties? Substantial size (littlest cycle length): for sharp move to waterfall locale; expansive least separation; Number of cycles of short length as little as could be expected under the circumstances; Very low blunder floors (bigness upgraded charts are not a decent decision Richardson 2003); Make the execution limit drawing nearer: sporadic degree dissemination (relies on upon the channel, "safe" hole to limit around 1dB); (Density Evolution, Richardson and Urbanke, 2001) Practical applications: numerical recipe for positions of ones in H

HOW CAN ONE SATISFY THESE CONSTRAINTS? For "standard channels" (symmetric, parallel information) the inquiries are for the most part surely knew BEC – thought to be a "shut case" (Capacity accomplishing degree dispersions, unpredictability of translating, halting sets, have straightforward EXIT diagram investigation… ) Error-floor most prominent staying unsolved issue! For "non-standard" channels still a considerable measure to take a shot at

CODE CONSTRUCTION AND IMPLEMENTATION FOR STANDARD CHANNELS

STRUCTURED LDPC CODES WITH GOOD PROPERTIES Structured LDPC: amiable for functional VLSI usage Construction issues: least separation, circumference/little number of short cycles, mistake floor properties Cycle-related imperatives ! Catching set related compelled

THE WATERFALL AND ERROR-FLOOR REGION Waterfall: Optimization of least separation and cycle length circulation (customary/unpredictable codes) Waterfall: Optimization of degree conveyance and code development Error floor: Optimization of catching sets or … possibly an alternate interpreting calculation? Reference book of Sparse Codes on Graphs: by D. MacKay http://www.inference.phy.cam.ac.uk/mackay/codes/data.html

Laendner/Leyba/Milenkovic/Prakash: Allerton 2oo3, 2004, ICC 2004, IT 2004, IT 2005; Tanner 2000; Vasic and Milenkovic 2001; Kim, Prepelitsa, Pless 2002; Fossorier 2004 Irregular Codes with vast bigness Masking… P

MATHEMATICAL CONSTRUCTIONS OF EXPONENTS P r 1 P r 2 P r 3 P r 4 c i c j c k c l LOW TO MODERATE RATE CODES: Fan, 2000 : Cycles exist if entirety of types in shut way is zero modulo measure ( P )= q Proper Array Codes (PAC) : Row-Labels { r i } frame an Arithmetic Progression with normal mean r i+1 - r i = an; Improper Array Codes (IAC) : Row-Labels { r i } don\'t shape an Arithmetic Progression ; 2 i + j - k - 2 l =0 i + j - k - l =0 2 k + i - 3 j =0 2 j - i - k =0 i + j - 2 k =0 mod q (CW3) i + j - 2 k =0 and i +2 j - 3 k =0 mod q (CW4)

ANALYTICAL RESULTS ON ACHIEVABLE RATES Number theoretic limits for contructions: For PACs with size g=8: For IACs with size g=10: Results in view of work of Bosznay, Erdos,Turan, Szemeredi, Bourgain s=1 0,1,5,14,25,57,88,122,198,257,280,... s=2 0,2,7,18,37,65,99,151,220,233,545,... s=3 0,3,7,18,31,50,105,145,186,230,289,... s=1 0,1,3,4,9,10,12,13,27,28,30,38,... s=2 0,2,3,5,9,11,12,14,27,29,30,39,... s=3 0,3,4,7,9,12,13,16,27,30,35,36,... Scientific developments: Joint work with N. Kashyap, Queen\'s , University, AMS Meeting, Chicago 2004

CYCLE-INVARIANT DIFFERENCE SETS Definition : V an added substance Abelian bunch, arrange v . ( v , c ,1) distinction set Ω in V is a c-subset of V with precisely one requested match ( x,y ) in Ω s.t. x-y = g, for a given g in V . Cases: {1,2,4} mod 7, {0,1,3,9} mod 13. "Inadequate" contrast sets – not each component secured CIDS: Definition Elements of Ω orchestrated in rising request. operator:cyclically shifts grouping i positions to the a good fit for i = 1,… , m, (part insightful) are requested contrast sets too Ω is a ( m+1)- overlap cycle invariant distinction set over V Example: V = Z 7 and Ω ={1,2,4}, m =2 mod 7

CIDS – GENERALIZATION OF BOSE METHOD where is the primitive component of GF ( q 2 ) , q odd prime S = built as Excellent execution for low rate codes

HIGH-RATE CODES: LATIN SQUARE CONSTRUCTION

ERROR FLOOR ANALYSIS Near codewords (Neal/MacKay, 2003) Trapping sets (Richardson, 2003) Considered to be the most remarkable issue in the zone! Innate property of both the disentangling calculation and code structure APPROACH: Try to foresee blunder floor (Richardson, 2003, 2004), or Try to change the translating calculation Observations prompting new interpreting calculation: Different PC codes give altogether different arrangements; quantization affects the mistake floor Certain factors in the code chart demonstrate to a great degree expansive and uncontrolled hops in probability values Strange things happen when message qualities are near +1/ - 1

OBJECT OF INVESTIGATION: MARGULIS CODE Elementary catching set (ETS): little subset of variable hubs for which actuated subgraph has not very many checks of odd degree INTUITION: channel commotion design with the end goal that all factors in ETS are off base after certain number of cycles; little number of odd degree checks shows that mistakes won\'t be effectively remedied Margulis code: in view of Cayley diagrams (arithmetical development), has great size Frame blunder rate bend has floor at 10E-7, SNR=2.4dB, AWGN Iteration number: 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 … . 182 142 108 73 66 55 38 29 20 16 15 14 … . Number of wrong bits: At "solidifying" point, all estimations of variable messages are +1/ - 1 and critical motions happen +1 → - 1 Frozen movement goes on for 13 emphasess: then "negative messages" proliferate through the entire diagram; can have bounced from - 0.33 to 49.67

ALGORITHM FOR ELIMINATING THE ERROR FLOOR "Motions" reminiscent to issues experienced in principle of disparate arrangement ( G.H. Solid, Divergent Series ) Trick: utilize a low-unpredictability "additional progression" in entirety item calculation in view of result by Hardy Parameters tunable: when do you "begin" utilizing "additional progression", what "numerical inclinations" should one utilize Can do thickness advancement investigation for enhancement purposes – no misfortune expected in waterfall district (nor one watched) (Paper with S. Laendner, in readiness) 1000 great edges: Standard and altered message passing both take 7.4 cycles by and large for right unraveling Modified message passing never takes more than 16 emphasess to meet 20 Bad edges (in MATLAB): Corrected 2/3 division of blunders after 35, 41, 26, 119, 38, 98… emphasess (standard fizzled even after 10000 cycles) Can utilize somewhat more confused result from G.H.H to lessen number of emphasis to 20-30

Check hub Variable hub VARIABLE AND CHECK NODE ARCHITECTURE (JOINT WORK WITH N. JAYAKUMAR, S. KHATRI) Globecom 2004, TCAS 2005 - Adder PLA From chec