1

Application of Nonbinary LDPC Codes Based onEuclidean Geometries to MIMO Systems1Xueqin Jiang, 1Yier Yan, 1,2Xiang-gen Xia Fellow and 1Moon Ho Lee Senior Member

1Institute of Information and CommunicationChonbuk National University, Korea.

2Department of Electrical EngineeringUniversity of Delaware, USA.

Abstract—This paper first presents an approach to the con-struction of non-binary low-density parity-check (LDPC) codesbased on Euclidean geometries. Codes constructed by this methodhave multiple code rates and a constant code length. With theproposed codes, the MIMO systems can also support differentdata rate with the same basic structure. Simulation resultsshow that these codes perform very well with either joint orseparate MIMO detection and channel decoding. The complexitycomparison shows that the nonbinary coded MIMO systems havelower complexity at the receiver side.

Index Terms—LDPC, Euclidean geometry, MIMO, FFT, FHT.

I. INTRODUCTION

Communication systems often need to work at differentdata transmission rates, which can be achieved by usingdifferent modulation schemes and/or multiple-rate codes. Tokeep the implementation as simple as possible, the same basicdecoder architecture should be able to decode the codes withdifferent code rates. The idea that higher rate quasi-cyclic(QC) effective low-density parity-check (LDPC) codes canbe generated from a lower rate QC mother LDPC code byrow combining in its parity check matrix was proposed in [1].The main issue that have to be considered in the constructionof the mother matrix and the effective matrices is that therows that will be combined in the mother matrix do not havenonzero elements in the same column. Codes of this familysupport different rates while maintaining the same fundamentaldecoder architecture.

The main contributions of this paper are summarized as: 1)We introduce an approach to the construction of multiple-ratenonbinery QC LDPC codes for the multiple-input multiple-output (MIMO) systems. Furthermore, we examine the per-formance of the proposed nonbinary LDPC codes for MIMOsystems that employ either joint detection and decoding (JDD)or separate detection and decoding (SDD) [2] and comparethem to the performance of the binary LDPC coded MIMOsystem in which the binary LDPC codes are from the IEEEstandard 802.16e [3]. 2) We analysis the complexities of thebinary and nonbinary LDPC coded MIMO systems. The pro-posed nonbinary LDPC MIMO system has lower complexitycompared to the binary LDPC MIMO system in which thebinary LDPC codes are from the IEEE standard 802.16e.

This paper is organized as follows. In Section II, we intro-duce the system model. Section III explains how to construct

the base matrix based on flats of two different dimensionsin Euclidean geometries. The method to get the mother andeffective matrices of multiple-rate is introduced in Section IV.Examples and their simulations are given in Section V. Thecomplexity analysis is presented in Section VI. Finally, sectionVII concludes the paper.

II. SYSTEM MODEL

Fig.1 shows a block diagram of the nonbinary LDPCcoded MIMO system. At the transmitter side, a sequence ofinformation bits is mapped to a sequence of nonbinary symbolsthrough a bit-to-symbol mapper. Let nt denote the number oftransmit antennas. At the output of the LDPC encoder, everygroup of n0 coded nonbinary symbols S = {s1, · · · , sn0} ∈GF (q) is mapped to a group of nt constellation symbolsx = (x1, · · · , xnt

) = f(S) through the mapper f(S). Giventhe constellation size 2m0 , we have p · n0 = nt · m0,where p = log2q. The sequence of constellation symbols issent through the nt transmit antennas. The receiver performsoptimal maximum a posteriori probability (MAP) detection tocompute the prior probabilities for each group of nt transmit-ted constellation symbols. These prior probabilities will thenbe passed to the LDPC decoder for iterative decoding.

When n0 = 1, the MAP detector produces prior probabili-ties for each symbol which can be used directly for nonbinaryLDPC decoding. Hence, it is sufficient to perform MIMOdetection only once followed by channel decoding. This cor-responds to SDD system that performs separate detection anddecoding. When n0 > 1, the prior probabilities of the group ofn0 nonbinary symbols are dependent because they are mappedto complex symbols that are transmitted simultaneously. Thenit is necessary to pass soft information about the dependentsymbols from the LDPC decoder back to the MAP detector toproduce updated symbol-wise probabilities. This correspondsto a JDD system that performs joint detection and decoding.

When q = 2, the block diagram shown in Fig.1 becomes abinary LDPC coded MIMO system. A sequence of informationbits is directly passed to a binary LDPC encoder. At the outputof the binary LDPC encoder, every group of n0 coded binarybits S = {b1, · · · , bn0} ∈ GF (2) is mapped to a group ofnt constellation symbols x = (x1, · · · , xt) = f(S). Given theconstellation size 2m0 , we have n0 = nt ·m0. It is obvious thatfor an MIMO system, nt > 1, the SDD system is impossiblefor binary LDPC codes.

978-1-4244-5668-0/09/$25.00 © 2009 IEEE

2

SymbolTo Bit

Bit ToSymbol

GF(q)LDPC

EncoderTransmiterf Receiver

MAPDetector

GF(q)LDPC

Decoder

1-f

JDD

Fig. 1. A schematic block diagram of the proposed nonbinary MIMO system.

The channel model is given by

y = Hcx + n, (1)

where x ∈ Cnt×1 is the complex transmitted signal vectorthat satisfies the component-wise energy constraint E(‖ xi ‖2

) = Es/nt, and Es is the total transmitted power, y ∈ Cnr×1

is the complex received signal vector, nr is the number ofreceive antennas, Hc ∈ Cnr×nt is the channel fading matrixwith independent entries that are complex Gaussian distributedwith zero mean and unit variance, n ∈ Cnr×1 is complexwhite Gaussian noise with variance σ2 per dimension. Hc isassumed to be known to the receiver but not the transmitter.

For a binary LDPC coded MIMO system, let Bi denote theset of (nt ×m0)−1 transmitted bits, but excludes the k-th bitbk while S(Bi, bk) is a vector of nt components containingthe symbols corresponding to the bit set of Bi and bk. Giveneach received signal vector y, we perform the bit based MAPdetection [4] to determine the a posteriori probabilities (APP)of each binary bit bk, k = 1, · · · , n0, by computing

PM→L(bk) =∑

all Bi

1(√

2πσ)nre

−‖y−Hcf(S(Bi,bk))‖2

(2σ2)

×∏

bj∈Bi

PL→M (bj) (2)

where ‖ ·‖2 denotes the norm square of a vector, the subscriptM → L denotes the message passed from the MIMO detectorto the LDPC decoder and L → M denotes the messagepassed from the LDPC decoder to the MIMO detector. Fora nonbinary LDPC coded MIMO system, let Si denote the setof nt − 1 transmitted symbols, but excludes the k-th symbolsk and S(Si, sk) is a vector of nt components containingthe symbols including Si and sk. Given each received signalvector y, we perform the symbol based MAP detection [4] todetermine the a posteriori probabilities (APP) of each symbolsk, k = 1, · · · , n0, by computing

PM→L(sk) =∑

all Si

1(√

2πσ)nre

−‖y−Hcf(S(Si,sk))‖2

(2σ2)

×∏

sj∈Si

PL→M (sj) (3)

Let P kM→L denote both of PM→L(bk) and PM→L(sk). Let

P kL→M denote both of PL→M (bk) and PL→M (sk). These

P kM→L values are passed to the FFT based LDPC decoder

for iterative decoding. The FFT with the correct symbol-bitlabeling reduces to the Fast Hadamard transform (FHT). Inthe iterative decoding algorithm, one iteration includes mainly

two steps [5]: 1) Variable node update for a degree dv node

Utp = P kM→L ×

dv∏v=1,v �=t

Vpv, (4)

where t = 1, ..., dv; 2) Check node update for a degree dc

node

Vtp = F−1(dc∏

c=1,c �=t

F (Upc)), (5)

where t = 1, ..., dc and F (·), F−1(·) denote the FHT andinverse FHT (IFHT), respectively. The extrinsic informationpass from the LDPC decoder to MIMO detector is

P kL→M =

dv∏v=1

Vpv. (6)

The permutations between the check nodes and variable nodesare omitted.

III. DESIGN OF THE BASE MATRIX B

Let EG(d, ps) be a d-dimensional Euclidean geometry overthe Galois field GF (ps), where p is a prime and d, s aretwo positive integers. Let c0, c1, . . . , cμ be μ + 1 linearlyindependent points in EG(d, ps), where 0 ≤ μ ≤ d. The pμs

points of the form c0 +β1c1 + . . . βμcμ with βi ∈ GF (ps) for1 ≤ i ≤ μ, constitute a μ-flat that passes through the pointc0. A point is a 0-flat and a line is a 1-flat. Let μ1, μ2 betwo integers and 0 ≤ μ1 < μ2 ≤ d, there are N(μ2, μ1, s, p)μ1-flats contained in a given μ2-flat and A(m,μ2, μ1, s, p) μ2-flats containing a given μ1-flat [6], where

N(μ2, μ1, s, p) = p(μ2−μ1)s

μ1∏i=1

p(μ2−i+1)s − 1p(μ1−i+1)s − 1

(7)

and

A(d, μ2, μ1, s, p) =μ2∏

i=μ1+1

p(d−i+1)s − 1p(μ2−i+1)s − 1

. (8)

The μ1-flats are ordered from 1 to N(d, μ1, s, p). Given a μ2-flat F and the incidence vector vF = (v1, v2, . . . , vN(d,μ1,s,p))of F be a binary N(d, μ1, s, p)-tuple with vi = 1 if the ith μ1-flat of EG(d, ps) is in F , and vi = 0 otherwise. The weightof vF is N(μ2, μ1, s, p). A μ-flat and the μ-flats parallel to itare called a parallel bundle. The μ-flats in a parallel bundleare parallel to each other.

The μ2-flats in EG(d, ps) can be partitioned into parallelbundles. The number of μ2-flats in a parallel bundle is p(d−μ2)s

and there are

K = N(d, μ2, s, p)/p(d−μ2)s =μ2∏i=1

p(d−i+1)s − 1p(μ2−i+1)s − 1

(9)

3

parallel bundles of μ2-flats. Denote these parallel bundles ofμ2-flats by P1, P2, . . . , PK . For each parallel bundle Pi forma p(d−μ2)s × N(d, μ1, s, p) incidence matrix bi whose rowsare the incidence vectors of the p(d−μ2)s μ2-flats in Pi overthe μ1-flats in EG(d, ps). Therefore, the rows and columnsof bi correspond to the μ2-flats in the parallel bundle Pi

and the μ1-flats in EG(d, ps), respectively. The μ2-flats ina parallel bundle are disjoint. Let b1, b2, . . . , bδ be δ incidencematrices of P1, P2, . . . , Pδ for 1 < δ ≤ K. Then we obtainthe following base matrix:

B =

⎛⎜⎜⎜⎝

b1

b2

...bδ

⎞⎟⎟⎟⎠ (10)

with N(d, μ1, s, p) columns and δp(d−μ2)s rows.

IV. MULTIPLE-RATE NONBINARY QC-LDPC CODES

A. Construction of The Mother Matrix

After replacing each ‘1’ entry in the m-th row and n-thcolumn of B with a shift value a(m,n) and replace ‘0’ elementwith ‘-1’. Then, the shift matrix

E(HM ) =

⎛⎜⎜⎜⎝

E(HM1)E(HM2)

...E(HMδ

)

⎞⎟⎟⎟⎠ (11)

is obtained, where a(m,n) means cyclic shift a square matrixto the right by a(m,n) positions. The mother matrix is obtainedby replacing each ‘-1’ element of E(HM ) with an L×L zeromatrix and replacing each a(m,n) with an L × L nonbinarycirculant permutation matrix whose shift value is a(m,n).In this paper we consider three classes of L × L circulantpermutation matrix over the nonzero elements of GF (q):1) L = q − 1. We replace each ‘-1’ of E(HM ) with a(q−1)×(q−1) zero matrix and replace each a(m,n) of E(HM )with a (q − 1) × (q − 1) α-multiplied circulant permutationmatrix. Each row of the α-multiplied circulant permutationmatrix is the right cyclic shift of the row above it multipliedby α and the first row is the right cyclic shift of the last rowmultiplied by α, where α is the primitive element of GF (q)[8][9][10]; 2)(q − 1) = kL. We replace each a(m,n) with a(q−1)/k×(q−1)/k β-multiplied circulant permutation matrix.Each row of the β-multiplied circulant permutation matrix isthe right cyclic shift of the row above it multiplied by β andthe first row is the right cyclic shift of the last row multipliedby β,where β = αk; 3)L = k(q − 1). We replace each a(m,n)

element of E(HM ) with a k(q − 1) × k(q − 1) α-multipliedcirculant permutation matrix. In this paper we simply call allof them the α-multiplied circulant permutation matrices. Themother matrix HM is

HM =

⎛⎜⎜⎜⎝

HM1

HM2

...HMδ

⎞⎟⎟⎟⎠ (12)

where HMicorresponds to bi and has p(d−μ2)s block rows

and N(d, μ1, s, p) block columns.Lemma 4.1: The block rows in HMi

do not have circulantpermutation matrices in the same block column.

Proof: Since the μ2-flats in each parallel bundle Pi aredisjoint (no points in common), no two incidence vectors oftwo μ2-flats in Pi can have any “1-component” in common.Therefore the rows in bi do not have ‘1’s in the same column.Consequently, the block rows in HMi

do not have circulantpermutation matrices in the same block column.

The null space of HM gives a mother code over GF (q)with design code rate

rM = 1 − δp(d−μ2)s

N(d, μ1, s, p)(13)

and code length L · N(d, μ1, s, p). The shift values assigningalgorithm in [7] is used in this paper to prevent short cyclesin the mother and effective matrices. The effective matriceswill be introduced in the next subsection. Since the effectivematrices have more short cycles than the mother matrix withthe row combining method, we only need to consider theeffective matrices when we assign the shift values in (11).

B. Construction of The Effective Matrices

For j = 1, 2, . . . , D, let HjE be an effective matrix obtained

by combining a number of block rows of HM . It is stated in[1] that the rows that will be combined in the mother matrixshould not have ‘1’s in the same column. From Lemma 4.1,we know that in each HMi

there is at most one circulantpermutation matrix in each block column.

Since the row weight of B is a constant numberN(μ2, μ1, s, p), the check nodes of HM have the same degree.Check node degree distribution for the effective code willbe concentrated if all the block rows in Hj

Eiresult from

combining the same number of block rows of HMi. Note

that the number of block rows of each HMiis p(d−μ2)s, we

can combine every pj block rows in HMito get one block

row in HjEi

. HjE has regular row weight pjN(μ2, μ1, s, p)

and therefore concentrated check node degree distribution. Theeffective matrix Hj

E is given by

HjE =

⎛⎜⎜⎜⎝

HjE1

HjE2...

HjEδ

⎞⎟⎟⎟⎠ (14)

where HjEi

corresponds to HMiand has p(d−μ2)s−j block

rows and N(d, μ1, s, p) block columns. Consequently, thenumber of possible effective matrices with different rates is

D = (d − μ2)s. (15)

The null space of HEigives an effective code over GF (q)

with design code rate

rj = 1 − δp(d−μ2)s−j

N(d, μ1, s, p)(16)

and code length L · N(d, μ1, s, p).

4

1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 610

−4

10−3

10−2

10−1

100

Eb/No (dB)

BE

R

802.16e r

M=1/2

802.16e r1=3/4

proposed rM

=1/2

proposed r1=3/4

Fig. 2. Simulation Results of Example 1

1 2 3 4 5 6 7 8 910

−4

10−3

10−2

10−1

100

Eb/No(dB)

BE

R

802.16e r

M=1/2

802.16e r1=3/4

proposed rM

=1/2

proposed r1=3/4

Fig. 3. Simulation Results of Example 2

V. SIMULATION RESULTS

Now, we will give two examples of our proposed codes andcompare them with the LDPC codes given in IEEE 802.16e.

Example 1: Consider the Euclidean geometry EG(4, 2).Let μ1 = 0, μ2 = 4, δ = 4 and L = 72. We construct a8×16 base matrix B. Consider the third class of α-multipliedcirculant permutation matrix introduced in the last section,since L = 24× (4− 1), we can replace each ‘0’ of B with an72 × 72 zero matrix and replace each ‘1’ with a 72 × 72 α-multiplied circulant permutation matrix over GF (4), then weobtain the a 576×1152 mother matrix HM over GF (4). Usingthe row combining method, we get effective matrices H1

E ofsize 288×1152. The null space of HM and H1

E give 4-ary QCLDPC codes of rate rM = 1/2 and r1 = 3/4, respectively.The binary code length is 2304. Every group of two codednonbinary symbols over GF (4) is mapped to a group of twoQPSK symbols and sent through two transmit antennas.

Example 2: Again consider the Euclidean geometryEG(4, 2). Let μ1 = 0, μ2 = 4, δ = 4 and L = 49.We construct a 8 × 16 base matrix B. Again consider thethird class of α-multiplied circulant permutation matrix, sinceL = 7 × (8 − 1), we replace each ‘0’ of B with an 49 × 49zero matrix and replace each ‘1’ with a 49× 49 α-multipliedcirculant permutation matrix over GF (8), then we obtain the

TABLE ISYSTEM SIMULATION PARAMETERS

No. of Transmitters 2No. of Receivers 2

Max No. of LDPC Iterations 25Max No. of Super Iterations 4

Channel Uncorrelated Rayleigh FadingCode Rates 1/2, 3/4

Galois Fields GF(4), GF(8)Modulation schemes QPSK, 8PSKSystem throughput 2,3,4.5 bits/symbol

a 392 × 784 mother matrix HM over GF (8). Using the rowcombining method, we get effective matrices H1

E of size192 × 768. The null space of HM and H1

E give 8-ary QCLDPC codes of rate rM = 1/2 and r1 = 3/4, respectively.The binary code length is 2352. Every group of two codednonbinary symbols over GF (8) is mapped to a group of two8PSK symbols and sent through two transmit antennas.

All the simulation parameters are listed in Table 1. Fromthese the simulation results of these two examples, we can seethat, MIMO systems coded by our proposed nonbinary LDPCcodes outperform the MIMO systems coded by the binaryLDPC codes given in IEEE 802.16e for the same modulationschemes, the same code rates and the same binary code length.

VI. COMPLEXITY ANALYSIS

As seen from (2), the decoding operations require theevaluation of all possible input symbol configurations contain-ing the k-th bit of {b1, · · · , bn0}, as well as the calculationof the a priori probability provided by the neighbouringbits. Hence, for a system having nt transmitters and m0

bits per constellation symbol, the computation of the term‖y − Hcf(S(Bi, bk))‖2 in (2) requires nt × nr multipli-cations, one multiplications for evaluating the square, andone for carrying out the required diving. One substraction isneeded for finding the Euclidean distance between the receivedsample and each of the constellation points. Furthermore,m0 × nt − 1 multiplications are needed for calculating theterm

∏bj∈Bi

PL→M (bj). 2m0nt−1−1 additions are needed forthe summation

∑all Bi

. There are 2m0nt values of S(Bi, bk).Thus for each decoded bit, the required number of multiplica-tions becomes ((m0 × nt − 1) + (nt × nr + 2)) × 2m0×nt =(nt × (m0 + nr) + 1) × 2m0×nt . The required number ofadditions is 2m0×nt + 2m0nt−1 − 1. For a nonbinary codedsystem using (3), the number of multiplications needed forthe apriori probability calculation is reduced to (nt − 1). Thenumber of additions used in the summation term is reducedto 2m0nt/2m0 −1. Thus the overall number of multiplicationsper bit is (nt(nr + 1) + 1) × 2m0×nt/m0 and the number ofadditions is 2m0×nt/m0 + (2m0(nt−1) − 1)/m0 per bit.

The well-known Bahl-Cocke-Jelinek-Raviv (BCJR) algo-rithm can be used for MAP decoding. The variable nodeand check node architectures are shown in Fig.4 and Fig.5,respectively. Assume one array multiplication include q mul-tiplications. As shown in Fig.4, the variable node needs3(dv −1) array multiplications. One more array multiplication

5

TABLE IICOMPLEXITY COMPARISON

Code Column Weight GF (q) Modulation Scheme No. of multiplications No. of additions802.16e rM = 1/2 3.125 GF (2) QPSK 165 37

First 802.16e r1 = 3/4 3.125 GF (2) QPSK 165 37Example proposed code rM = 1/2 2 GF (4) QPSK 163 56

proposed code r1 = 3/4 2 GF (4) QPSK 163 56802.16e rM = 1/2 3.125 GF (2) 8PSK 725 109

Second 802.16e rM = 3/4 3.125 GF (2) 8PSK 725 109Example proposed code rM = 1/2 2 GF (8) 8PSK 168 56

proposed code r1 = 3/4 2 GF (8) 8PSK 168 56

L

1 pU

tpU

vdpU

1pV

pvV

vpdV

Uq

q

q

q

q

q

q

q

Backward Forward Combination

Fig. 4. Variable Node Architecture

FHT FHT FHT FHT FHT

IFHT IFHT IFHT IFHTIFHT

V V V VV

UpcUU U U

cpd3p2p1p

cdptp3 p2 p1 p

Forward

Backward

Conbination

Fig. 5. Check Node Architecture with FHT and IFHT

is needed to calculate U which is the output of the decodinghard decision. Therefore, one variable node of degree dv

needs 3qdv −2q multiplications. As shown in Fig.5, the checknode needs 3(dc − 2) array multiplications, dc FHTs and dc

IFHTs. A q-point FHT or IFHT needs q log2 q additions. Onecheck node of degree dc needs 3q(dc − 2) multiplicationsand 2dcq log2 q additions. The number of multiplications andadditions required for each decoding bit can represented as(3qdv − 2q)/p + 3q(dc − 2)/(p × dc) ≈ q(3dv + 1)/p and2dvq log2 q/p, respectively. Note that the binary LDPC codesis a special case of the LDPC codes over GF (q) (q=2).Therefore the above formulas for the number of multiplicationsand additions can also be used for the binary LDPC codes.

For a binary LDPC coded MIMO system, the number ofmultiplications required for decoding each bit, in one superiteration, is (nt × (m0 + nr) + 1) × 2m0×nt + 6dv + 2 andthe number of additions required is 2m0×nt +2m0nt−1 +4dv .For the nonbinary LDPC coded MIMO system, the numberof multiplications required for decoding each bit, in one superiteration, is (nt×(nr+1)+1)×2m0×nt/m0+q(3dv+1)/p andthe number of additions required is 2m0×nt/m0+(2m0(nt−1)−1)/m0+2dvq. The complexities of the systems given in Table

I are listed in Table II.

VII. CONCLUSION

In this paper, we proposed a class of multiple-rate nonbinaryLDPC codes for MIMO systems. These codes of multiple-ratecan be supported by the same decoder architecture. Therefore,the MIMO system with this code can support different datatransmission rates with the same basic structure. Furthermore,MIMO systems coded by the proposed codes have lowercomplexities and better performance than the systems codedby the binary LDPC codes given in IEEE 802.16e.

ACKNOWLEDGMENT

This work was supported by World Class University R32-2008-000-20014-0 NRF, and KRF-2007-521-D0030, Korea.

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