Design of Non-Binary Error-Correction Codes and their Applications R. A. Carrasco and. Johnston, Non-Binary Error Control Coding for Wireless Communication and Data Storage, Wiley, SBN 978-- 7-89-9 Prof. R. A. Carrasco Cork 9
Outlines Non-binary DPC codes Construction methods Complexity reducing decoding Performance evaluation Ring-TC codes Encoder structure Advantage discussion Application in cooperative systems Algebraic-Geometric codes Sponsored by with EP/E8/ Sponsored by with EP/E8/ Code construction Decoding systems and complexity reduction Application to storage devices Sponsored by with EP/D88
Non-binary DPC code construction methods H = v v v v v v v7 v8 z z z z The marked path v z v z v is a cycle of length four in the Tanner graph. Short cycles should be avoided since they can prevent optimal decoding. The minimum length cycle in a Tanner graph is known as the girth of the code Key issue: Non-binary DPC code construction methods should ensure the codes are free of short cycles (cycles of length four) and maximise the girth.
Non-binary DPC code construction methods Random construction: A binary parity check matrix is constructed at random, having constant column weight and constant row weight. The short cycles are then eliminated using exhaustive computer search. The binary elements are then replaced by carefully selective non- binary element of GF(q). Regular construction: Example: H = Each row has row weight w r and each column has column weight w c. No two rows or two columns have non-zero elements in the same position. This ensures that the code is free of cycles of length four (short cycles). Quasi-Cyclic (QC)-DPC codes QC-DPC codes constructed from Reed-Solomon codes
Non binary QC DPC Codes Non-binary QC-DPC Codes A circulant matrix of size q x q is defined as the identity matrix where each row is multiplied by an increasing power of and where each row is multiplied by an increasing power of and GF(q). = Circulant matrices are placed within the overall parity check matrix to form regular QC- DPC codes. DPC codes. et s and t be two non zero elements with multiplicative orders O(s)=k and O(t)=j. The order of an element s in a finite field is the smallest integer v > such that s v =. We could construct a k x j based matrix B as: j =... k k t s t s s t t s s s B K K... j k j j j t s t s s t t K K K K K K
N bi QC DPC d The QC-DPC code is constructed by replacing each element in B with li ll hift d ( ) i l t t i Non-binary QC-DPC codes cyclically shifted (q q) circulant matrix. B = B = Specify the number of cyclic shifts to The overall dimension of the parity check matrix H of size (jq) (kq) is k s s s K p y (jq) ( q) constructed as: =..... k t s t s t s t q k q j H K K K K K K... j k j j j t s t s t s t K
Non-binary QC-DPC codes Non-binary QC-DPC codes We can produce the top left- Given q = 7, k = and j =, = H p p hand corner of the overall nonbinary parity check matrix H as shown below H =
Non-Binary QC-PDC codes constructed from Reed-Solomon codes in et al demonstrated how non-binary DPC codes can be constructed from a (n,, n-) Reed-Solomon code defined over GF(q) with two information symbols. k is the message length and d is the minimum Hamming distance. Given two codewords c = [,,,..., ] and c = [,,,..., q- ], since Reed-Solomon codes are linear codes, then codeword c = c + c = [, +, +,..., + q- ] Codeword of row weight n- We can form the rows of a base matrix B by taking c and cyclically y shifting it for each row B = q + + + + + + + q + + q Weight n Reed-Solomon codeword
Non-Binary QC-PDC codes y constructed from Reed-Solomon codes Each element i in the base matrix B is then replaced with a different type of (q ) (q ) non-binary circulant matrix Z( i ) = [z( i ), z( i ),., z( q- i )] T. et z( i ) = [z, z, z,..., z q- ] is a vector defined over GF(q), with element z i = i GF(q) and all other elements are zero. z( v. i ) corresponds to v right cyclic shift of z( i ) multiplied by v. Example: Given z( ) = [,,,,,, ], then matrix Z( ) is: z( ) = ) ( Z z(. )
Non-Binary QC-PDC codes constructed from Reed-Solomon codes Non-binary circulant matrix Given c = [,,,,,, ] and the base matrix B B = Substitute matrices Z(), Z( ) Z( ) O Z( ), Z( ) and Z()
Reducing the Complexity of the non- binary DPC Decoder The Belief Propagation (BP) algorithm for the decoding of non-binary DPC codes is too complex, particularly for high order finite fields., ost of the complexity is due to the Horizontal step of the BP algorithm By recognising that this step is a convolution operation we can replace it using a simple Fast Fourier Transform (FFT).
Belief Propagation (BP) Algorithm Take a simple regular binary parity check matrix H with row weight of and column weight of The BP algorithm Coded bits z =c c c nitialisation H = c c c c c c z Horizontal step: r mn (x) Parity z Check ( ) Equations Vertical step: q mn(x) z ĉ z Syndrome check: S = H ĉ f S f S = c = ĉ
Probabilities associated with BP Decoding The Tanner graph for H c c c c c c Horizontal step r mn ( x) = c: c n = x P( z m = c) n' N m q mn' \ n ( x) where N m is the set of coded bits connected to the parity check z m Vertical step q r q x r q = β mn ( x) mn f n z z z z m' n rm ' n ( x) \ m where β mn is a normalising constant, f nx is the likelihood of the n th received bit being equal to x GF(q) and n is the set of parity checks connected to c n. q mn (x) is the probability that the n th coded bit c n = x, given the values of the parity checks connected to c n excluding the m th parity check z m r mn (x) is the probability that the m th parity check z m =, given the values of the coded bits connected to z m and with the n th coded bit c n = x n
The Horizontal Step viewed as a convolution operation The Horizontal Step determines the probabilities r mn (x). r mn ( x) = P( zm = c) qmn' ( x) c: c = x n' N n n m \ Consider the first row of H, z = c c c. r () is the probability that the first parity check z = given the values of the coded bits connected to it and with c =. i.e. c = c c = The possible values of c and c that satisfy this are: c =, c = and c =, c = Therefore, r () = q ().q () + q ().q () r () can then be found using r () = r () Thiscanbe rittenas written r ( x ) = q ( v ) q ( x v), which h is a discrete convolution ) v=
The BP algorithm for non-binary DPC codes The diagram below shows the BP algorithm for non-binary DPC codes. f i is the received symbol likelihood, h ij are the non-binary elements of H and Π shows the connections between the coded bits and the parity checks. f f f n c c q mn c n De-permuted r mn h h h m h h h m h n h n h Permuted mn q h c h c h m c h c h c h m c mn h n c n h n c n h mn c n r mn CONV CONV CONV z z z m
Replacing the Convolution Operation with FFTs f f f n Permute c c q mn c n De-permuted r mn De-permute FFT h h h m h h h m h n h n h mn Permuted q h c h c h m c h c h c h m c mn h n c n h n c n h mn c n F F F F F F F F F r mn nverse FFT z z z m
Complexity Reduction The complexity of the Horizontal Step of the BP algorithm can be shown to be O(q ), where q is the cardinality of the finite field. Since the FFT step is analogous to the Cooley-Tukey algorithm, the complexity of the Horizontal step is reduced to O(q.log (q))
Simulation Results for non-binary DPC codes from the Reed-Solomon Construction ethod..e+.e-.e- BER Uncoded AWGN ( trials) GF(), N= GF(), N= GF(), N= GF(), N=78 GF(), N= BER.E-.E-.E-.E- (Eb/N) SNR db Non-binary DPC codes defined in a higher order finite field achieve a better performance on AWGN channel.
Cooperative Communication with Non- Binary DPC codes Cooperative signal model Cooperative transmission could be described by a two-phase process n the first phase (nitial transmission) S x S x S y SD D y SD[n] = a SD x s [n] + z SD[n] y SR[ [n] = ε a SR x s s[ [n] + z SR[ [n] y SR n =,,, N R n the second phase (relaying transmission) R D n =,,, N y RD x R ε y RD [n] = ε a RD x R [n] + z RD [n] Source to destination transmission Source to relay transmission Relay to destination transmission After the two phases: destination will combine the signal from source y SD and relay y RD in the two phases
Cooperative Communication with Non-Binary DPC codes Cooperative protocols: Amplify-and-Forward (AF): relay amplifies the received signal of the initial transmission and forwards to destination; Decode-and-Forward d d (DF): relay decodes d the received signal. f decoding di is successful, it will re-encode it and forward it to the destination; Cooperation achieved through h Time Division i i Channel Allocation Direct Transmission S Tx R Tx Cooperative Transmission S Tx R Tx for S R Tx S Tx for R Non-binary DPC code is applied in the DF cooperative model with using complexity reduction decoding;
Performance of Cooperative Communications with Non- Binary QC-DPC Codes Constructed from Reed-Solomon Codes Performance of the same DPC codes for interuser channel SNR of db.e+.e- gf() DF - db interuser Channel gf() g( DF - db interuser Channel GF() DF - db interuser Channel.E- BER.E-.E-.E- 7 8 9 SNR db Non-binary DPC codes defined in a higher order finite field achieves a better performance in cooperative DF channel.
Ring-TC Codes Encoder structure m m c c m k g s k g g s s g k g g g k g g c k c k+ f s f essage symbols m, m,, m k, tap coefficients g i, g i,, g ik (i =,,, s), f,, f s, and output coded symbols c, c,, c k, c k+ are Z q elements {,,,, q-}; ength of message symbols k, length of coded symbols k+, number of memory elements s; c, c,, c k, c k+ are directly mapped to a Gray-coded q-psk symbol Code indications: Ring-TC (g sk,, g s, g s ) (g k,, g, g ) (g k,, g, g ) / (f s,, f ) qpsk.
Ring-TC codes Advantages: No need for doing set partitioning just using Gray-coded qpsk; Easy to generate linear codes over rings, they are distance invariant, decision region invariant and mostly transparent to multiples of π/q phase rotation; Using less memory units to generate a code with high number of states; Achieve better asymptotic coding gain than PSK & 8PSK TC codes; ore importantly Compared with other trellis code, it has the optimised performance over slow Rayleigh fading channel; t also has the high information throughput feature makes it the ideal coding scheme to be applied in cooperative systems. Note: For further information about Ring-TC code design criterion and space-time Ring-TC code, refer to the book Non-binary error control coding for wireless communication and data storage.
Ring-TC Codes for cooperative systems Signal model for cooperative Amplify-and-Forward (AF) scheme nitial transmission (st Time Slot) Relaying transmission (nd Time Slot) S x S y SD D D x S y SR R x R R y RD y SD [n] = ε a SD x s [n] + z SD [n] y SR [n] = ε a SR x s [n] + z SR [n] n =,,, N x R [n] = β y SR [n]; Amplification gain β = ( a SR ε + N, SR ) - ; y RD [n] = ε a RD x R [n][ ] + z RD [n][ ] n =,,, N After two Time Slots, received symbols are combined using aximal ikelihood detection ti as: r SD [n] = w SD y SD [n] + w RD y RD [n] * asd ε wsd = N, SD w RD ' = a RD * SR a βa β N * RD, SR ε + N, RD Combining gains
Ring-TC Codes for cooperative systems Spectral efficiency analysis for the AF scheme Time-division channel allocation: Direct Transmission ST Tx RT Tx Cooperative Transmission S Tx R Tx for S R Tx S Tx for R Each user sacrifices half of its transmission freedom (Time) for relaying other s signal; Define the spectral efficiency of a user as: = number of its info bits / each of its transmitted symbol (info bits/symbol, or bits/s/hz); n a communication system employing a code with rate R and modulation scheme with order ( bits/symbol), then dir = R info bits/symbol; coop = R/ info bits/symbol;
Ring-TC Codes for cooperative systems dir = R info bits/symbol; coop = R/ info bits/symbol; To compensate this loss, and achieve dir = coop, one might have to increase modulation scheme order or code rate R; Work done by our Cambridge colleagues: Direct: rate / conv (, 7) 8 with QPSK, dir = info bits/symbol; Coop: rate / conv (, 7) 8 with QA, coop = info bits/symbol; Only employing a high-order modulation scheme to compensate the spectral efficiency loss is on the expense of severe performance degradation; Consider both the code rate R and modulation scheme together Ring- Trellis Coded odulation (TC)!
Ring-TC Codes for cooperative systems Comparing Ring-TC TC(, )(, )/() 8PSK Conv (, 7) 8 QA Both codes have states, achieve spectral efficiency of coop = bits/symbol Ring-TC (, ) (, ) / () 8PSK Conv (, 7) 8 QA c c c ap pping 8PSK symbols QA symbols
Ring-TC Codes for cooperative systems Performance evaluation of symmetric uplink scenario S D.E+ SNR SD = SNR RD nteruser channel R.E- Uplink channels.e- BER.E-.E-.E-.E- Direct Ring-TC 8PSK (db interuser channel) Conv QA (db interuser channel) Ring-TC 8PSK (db interuser channel) Conv QA (db interuser channel) Ring-TC 8PSK (db interuser channel) Conv QA (db interuser channel) Ring-TC 8PSK (Perfect interuser channel) Conv QA (Perfect interuser channel) 8 8 8 SNR of Uplink Channels, Es/N (db) Ring-TC code outperforms the convolutional code; Diversity gain could be achieved with at least db interuser channel
Algebraic-Geometric Codes Currently, Reed-Solomon codes are used in magnetic and optical storage devices. As storage densities increase so will the effects of inter-symbol interference (S) and a more powerful error-correcting code will need to replace them. Algebraic-geometric (AG) codes are a class of non-binary block codes presented by Goppa in 98 constructed from the affine points and points at infinity of an irreducible smooth projective curve. AG codes are longer than Reed-Solomon codes since there are more affine points on a curve than a line over the same finite field. Hence, they have larger minimum Hamming distances. Example, defined over GF( 8 ), the RS(,, ) code could correct burst errors of up to bits; The Hermitian(9,, ) code could correct burst errors of up to 9 bits (just over times longer!). There is also a greater availability of AG codes since there are many different classes of curves. e.g. Hermitian curve, Elliptic curve. Decoding complexity is higher h than the decoding di complexity of Reed-Solomon codes
Construction of a Hermitian code Hermitian codes are a class of algebraic-geometric codes constructed from the points on a Hermitian curve defined over GF(r ) given as r+ C ( x, y, z ) = x + y z + r yz r The curve has r affine points and one point at infinity. Associated with each curve is a parameter known as the genus γ, which h for Hermitian curves is r( r ) γ = The codeword length n, message length k and designed minimum distance d* of a Hermitian code are n = r d * k = n a + γ = a γ + where γ < a < n
Construction of a Hermitian code A generator matrix G of a Hermitian code is constructed by evaluating a set of k A generator matrix G of a Hermitian code is constructed by evaluating a set of k rational functions, where k is the dimension of the code, with increasing pole orders up to a desired value at each of the r affine points. This set of rational functions is fth f of the form a j r ri z y x f f f aq j i j i k + + = = + ) (, },,, { ) ( H G i d fi d A d d d C i t d Hence, G is defined as And codeword C is generated as: ) ( ) ( ) ( ) ( ) ( ) ( r r P f P f P f P f P f P f G C = m G = ) ( ) ( ) ( ) ( ) ( ) ( r k k k r P f P f P f f f f O G where m is the message vector m = (m, m,, m k ) Alternatively, by defining the message polynomial f as: f = m f + m f + + m k f k the codeword C = (c, c,, c n ) (n = r ) is generated as; (,,, n) ( ) g ; (c, c,, c n ) = (f(p ), f(p ),, f(p n ))
Decoding algorithms The Sakata algorithm with majority voting hard-decision unique decoding resulting a unique decoded results; efficient but error-correction capability is limited by the half distance bound; The Guruswami-Sudan algorithm hard-decision list decoding resulting a list of decoded results; error-correction can exceed the half-distance bound, but with high complexity; Complexity reduction can be achieving by eliminating unnecessary polynomials during interpolation; The Koetter-Vardy algorithm soft-decision list decoding soft-decision version of the Guruswami-Sudan algorithm; exceed the Guruswami-Sudan algorithm s optimal bound with less decoding complexity; Note: A detailed description of the decoding algorithms is included in the book Non-binary error control coding for wireless communication and data storage.
The Sakata algorithm with ajority voting Decoding process: m Algebraic Geometric Systematic Encoder c odulation Fading Amplitude Gaussian Noise Demodulator r CHANNE AGEBRAC-GEOETRC DECODER PROCESS $m essage Extraction * d Decoding boundary: τ = $c nverse Discrete Fourier Transform Error agnitudes Unknown syndrome j ajority Known syndrome Voting Sakata Algorithm Error ocation Find known syndromes
The Guruswami-Sudan algorithm Decoding process: given hard-decision received word R = (R, R,, R n ) nterpolation Factorisation R Q(x, y, z) h terative polynomial construction Recursive coefficient search Complexity is dominated by this process. Complexity could be reduced by eliminating unnecessary polynomials! Output candidates to be the message polynomial nterpolation: terative modification for a group of trivariate polynomials. The output polynomial Q(x, y, z) has a zero of multiplicity m over all the interpolated units (P, R ), (P, R ),, (P n, R n ). Factorisation: Recursive coefficient search process to determine the message polynomial s coefficients. Decoding boundary: * τ = n n( n d )
The Koetter-Vardy algorithm Based on the received word R, a reliability matrix π of size q n is generated. ts entry π i,j = Prob(ρ j was transmitted r i is received); Reliability matrix π is then transformed to a multiplicity matrix ; Decoding process: π nterpolation Q(x, y, z) Factorisation h terative polynomial construction Recursive coefficient search For interpolation, each nonzero entry m i,j of indicate an interpolated unit (P j, ρ j ). * Decoding boundary can well exceed τ = n n( n d )
Performance comparison of the three decoding algorithms BER.E+.E- E.E-.E- Hermitian code (, 89) over AWGN channel Uncoded Sakata hard-decision (Optimal) soft-decision (l = ) soft-decision (l = ) soft-decision (l = ) soft-decision (l = ) soft-decision (l = ) soft-decision (l = ) soft-decision (Optimal) l is the output list size of the list decoder.e-.e-.e- 7 E b / N [db] Coding gains of the KV algorithm over the GS algorithm! Coding gain of the GS algorithm over the Sakata algorithm!
Application of Hermitian Codes to agnetic Storage Devices nput {, } Block Encoder Σ Σ S AWGN f f f D D Σ Partial Response Channel Partial response polynomials of the form f(d) = f + f D + f D +..., where f i R can be used to model a perfectly equalised magnetic storage channel For a longitudinal magnetic channel polynomials of the form f(d) = ( D)(+D) p are commonly used, where p is an integer. When p =, f(d) = D and is called PR For a perpendicular magnetic channel polynomials of the form f(d) = ( + D) p are commonly used. When p =, f(d) = + D + D and is called PR
Application of Hermitian Codes to agnetic Storage Devices AP Detector Decoder Decoded Data The partial response channel takes a binary input and gives a multi-level output. The output of the map detector gives the likelihoods of the read bits A hard-decision on the soft output is made and the resultant bits are mapped to symbols in the finite field for the Hermitian code decoder f a soft-decision decoder is used, such as the Belief Propagation algorithm for DPC codes, then the bit likelihoods from the AP detector are directly used.
Simulation Results of Hermitian Code on PR Channel.E+.E-.E- Uncoded PR (, 8, 99) Hermitian code (, 99) binary DPC code BE ER.E-.E-.E- Hermitian code outperforms the DPC code at BER. -.E-.E-7 7 7 8 9 7 8 9 SNR, db A comparison of the ( 8 99) Hermitian code in GF() with the ( 99) A comparison of the (, 8, 99) Hermitian code in GF() with the (, 99) binary DPC code, column weight row weight 8, on the PR channel with AWGN.
Simulation Results of Hermitian Code on PR Channel E+.E+.E-.E- Uncoded PR (,8,99) Hermitian code (,99) binary DPC code BE ER.E-.E-.E- Hermitian code outperforms the DPC code at BER = -.E-.E-7 7 8 9 7 8 SNR, db A comparison of the (, 8, 99) Hermitian code in GF() with the (, 99) binary DPC code, column weight row weight 8, on the PR channel with AWGN.
Conclusions Non-binary DPC Codes Construction methods of non-binary DPC codes QC non-binary DPC codes and QC non-binary DPC codes constructed from Reed-Solomon codes; Efficient i decoding di for non-binary DPC codes replacing the convolutional l process of Horizontal step by the FFT; Ring-TC Codes Construction methods and its application to cooperative systems; t not only achieves high spectral efficiency, but also optimised performance in cooperative systems; Algebraic-Geometric Codes Construction methods for algebraic-geometric codes; Decoding algorithms for AG codes with complexity reduction method; ts application to magnetic storage channel it could outperform DPC codes;