Multilevel Codes in Lattice-Reduction-Aided Equalization. Author(s): Fischer, Robert F.H.; Huber, Johannes B.; Stern, Sebastian; Guter, Paulus M.

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1 Research Coection Conference Paper Mutieve Codes in Lattice-Reduction-Aided Equaization Author(s): Fischer, Robert FH; Huber, Johannes B; Stern, Sebastian; Guter, Pauus M Pubication Date: Permanent Link: Rights / License: In Copyright - Non-Commercia Use Permitted This page was generated automaticay upon downoad from the ETH Zurich Research Coection For more information pease consut the Terms of use ETH Library

2 Internationa Zurich Seminar on Information and Communication (IZS), February 21 23, 2018 Mutieve Codes in Lattice-Reduction-Aided Equaization Robert FH Fischer, 1 Johannes B Huber, 2 Sebastian Stern, 1 Pauus M Guter 2 1 Institut für Nachrichtentechnik, Universität Um, Um, Germany, Emai: {robertfischer,sebastianstern}@uni-umde 2 Lehrstuh für Informationsübertragung, Universität Erangen-Nürnberg, Erangen, Germany, Emai: johanneshuber@faude Abstract The appication of mutieve codes in attice-reduction-aided equaization is considered, ie, the decoding of mutieve codes when (Gaussian) integer inear combinations of the codewords in signa space are present Typicay, mutieve codes do not generate attice codes, hence arbitrary integer inear combinations are not directy decodabe We show that this attice property is not required, which reaxes the constraints on the component codes significanty A generaized version of mutistage decoding which incorporates a carry correction is proposed; it circumvents the attice property and any integer inear combinations are decodabe Numerica simuations are given to cover the performance of the proposed method I INTRODUCTION In the iterature, ow-compexity but we-performing approaches for the equaization in mutipe-input/mutipe-output (MIMO) muti-user upink scenarios are discussed for more than one decade Lattice-reduction-aided (LRA) techniques [16], [15] and the tighty reated concept of integer-forcing (IF) receivers [10], [17] are of specia interest In both schemes, the main idea is not to decode the transmitted signas, but integer inear combinations thereof; LRA and IF receivers differ in the way the (residua) integer interference is handed, cf [4] Right from the start, IF schemes were proposed as coded schemes a strong couping between integer equaization and decoding/code constraints is present In contrast, LRA schemes are usuay treated uncoded as a combination with coded moduation schemes seems to be easier Here, the actua demand is that the code is inear in signa space; any integer inear combination of codewords has to be a codeword Lattice codes obviousy fufi this demand In this paper, we are interested in the combination of mutieve codes (MLC) and LRA equaization, because mutieve coding together with mutistage decoding (MSD) is in principe a capacity-achieving strategy [14] Typicay, mutieve codes do not generate attice codes [9] and are not inear However, we show that this property is indeed not required We study the reevant design criteria on the component codes and propose a generaized version of mutistage decoding which incorporates a carry correction such that integer inear combinations are decodabe 1 The paper is organized as foows: In Sec II the system mode is introduced The consequences of integer inear combinations on mutieve codes are studied in Sec III and a new decoding scheme with carry correction is proposed Sec IV presents numerica exampes; the paper is briefy summarized in Sec V II SYSTEM MODEL Throughout the paper, we assume K non-cooperating (singeantenna) users k, k = 1,, K, communicating their binary source symbos 2 q k F 2 to a centra receiver with N R K antennas To that end, the symbos are encoded and mapped to compex-vaued transmit symbos x k, drawn from some signa consteation A with variance σx 2 Denoting the K-dimensiona transmit vector as x, the N R K (fat-fading) channe matrix as H, and the N R - dimensiona noise vector (with zero-mean Gaussian noise components with variance σn 2 per dimension) as n, the receive vector y (compex baseband notation) reads as usua y = Hx + n (1) We assume joint processing of the N R components of the receive vector y The common, practicabe approach is to first perform some form of joint equaization, foowed by individua decoding of the codes Lattice-reduction-aided and integer-forcing equaization are ow-compexity, we-performing approaches In both variants, the main idea is to factorize the channe matrix as H = W Z, (2) where Z G K K, G = Z + jz, is a fu-rank (Gaussian) integer matrix Then, ony the non-integer part W is equaized (here via MMSE inear equaization) [3], [17], [4] The different factorization criteria (attice or dua-attice approach, ZF or MMSE criterion), constraints (unimoduar vs fu-rank), and agorithms (shortest basis probem vs shortest independent vector probem) are irreevant for the scope of the paper; for an overview see, eg, [4] Incuding the inear (ZF or MMSE) equaizer frontend, the remaining part of the (LRA/IF) receiver has to dea with r = Zx + n = def x + n, (3) where n is the effective disturbance after equaization here, for brevity we assume that a (compex-vaued) components have the same variance σ 2 n Fig 1 shows the effective end-toend channe mode The main difference between LRA and IF equaization is how the integer interference is resoved In LRA schemes, the 1 Shorty after the submission of this paper we became aware of a simiar work [1], however, treating ony coding over ASK consteations 2 The notation distinguishes quantities over the compex numbers (typeset as x, Z, ), and over finite fieds (typeset in Fraktur font; q, c, Z 0, ) 133

3 Internationa Zurich Seminar on Information and Communication (IZS), February 21 23, 2018 q 1 q K F 2 ENC / M ENC / M x 1 x K C Fig 1 End-to-end integer channe mode (incuding inear equaization) LRA IF r r C ˆ x 1 ˆ x K ˆ x 1 ˆ x K Z Z 1 ˆ q 1 ˆ q K F p ˆx C Z 1 n Fig 2 System mode of the receiver over end-to-end integer channes Top: attice-reduction-aided equaization; Bottom: integer-forcing receiver inear combinations x of the transmit symbos in signa space are estimated by the decoders and the interference is undone via Z 1 over the compex numbers, cf Fig 2, top Finay, via the encoder inverses, estimates ˆq k are produced In IF schemes (Fig 2, bottom), the source symbos are drawn from a finite fied F p, p a prime, and encoding is done over F p At the receiver, inear combinations q k of the source symbos are deivered by the decoders and the integer matrix Z (the F p -equivaent to Z) is inverted over F p These different orders of encoder inverse and inverse of Z impose different constraints on the codes In LRA schemes, integer inear combinations in signa space have to be decodabe When attice codes are used, this is directy fufied III MULTILEVEL CODES AND INTEGER INTERFERENCE A particuar strategy to coded moduation is mutieve coding [7], [14] Via a set of binary component codes C, = 0,, m 1, and a mapping from binary address abes of m = og 2 (M) digits to M signa points (constituent consteation), a code in signa space is generated A Mapping We are interested in one-dimensiona (ampitude-shift keying, ASK) and (square) two-dimensiona (quadrature-ampitude moduation, QAM) constituent consteations and restrict ourseves to mapping according to the set partitioning rue [13] In both cases, the mapping can be written as [5] M(b m 1 b 1 b 0 ) = mod B ( m 1 r ˆq 1 ˆq K F 2 =0 ψ(b )φ ) O, (4) where ψ( ) is the common mapping from the finite fied (F 2 ) eements 0 and 1 to integers 0 and 1, ie, ψ(0) = 0 and ψ(1) = 1 B = M for ASK and B = M for QAM ˆq defines the boundary region 3 and O = M 1 2 for ASK and O = (1 + j) M 1 2 for QAM is the offset for zero-mean consteations Ignoring the moduo reduction and the offset, the point in signa space is given by its binary expansion wrt the base φ For ASK we have φ = 2 and the usua binary expansion of an integer is present For QAM we choose φ = 1 + j This is due to the fact that the Gaussian integers G can be uniquey given via a binary representation where the base (radix) is the compex number ±1 ± j [11], [6], cf aso [5] B Mutieve Codes and Lattices Using this mapping, the mutieve code is defined by C MLC = mod B ( m 1 =0 ψ(c )φ ) O, (5) where ψ( ), mod( ), and the offset O are appied componentwise and C, = 0,, m 1, are the binary component codes (at eve ) Here, to simpify exposition, they are assumed to be bock codes of equa ength N Such a mutieve code is never a attice However, we can consider the foowing construction ( construction by code formua [9], [5]) C = m 1 ψ(c )φ + φ m G N, (6) =0 which differs from C MLC by i) eiminating the offset O and ii) inherenty assuming an infinite number of uncoded eves via the addition of φ m G N (φ m Z N in case of ASK), ie, periodic extension by adding integer mutipes of φ m to the coordinates of the codewords (note that φ m = B is the size of the support per dimension of the mutieve code) In [9] it is shown that C is a (rea) attice, ie, has group structure under ordinary vector addition (over R), if a) C are inear codes b) C 0 C 1 C m 1 (7) c) for a and c i, c j C it has to hod c i c j C +1 where denotes the eement-wise mutipication (in the arithmetic of F 2, ie, AND operation) of the words c i and c j The intuition behind the third demand is that the addition of two mapped words in rea space (ASK) can be written as [9] ( : eement-wise addition in the arithmetic of F 2 ) ψ(c i ) + ψ(c j ) = ψ(c i c j ) + 2ψ(c i c j ), (8) where c i c j is the carry since = 2 = [1 0] 2 For QAM signaing we have to resort to the arithmetic wrt the base 1 + j Here, 2 = [ ] 1+j which gives ψ(c i ) + ψ(c j ) = ψ(c i c j ) + (φ 2 + φ 3 )ψ(c i c j ), (9) ie, the carry is 110 [6] and c i c j C +2 C +3 is the respective constraint ((7)c) is sufficient but not required) 3 mod B (x) 0,, B 1 if x is rea-vaued; mod is appied separatey to rea and imaginary part when x is compex-vaued 4 The sum of two codewords in signa space is ony a vaid codeword if the sum of two codewords in Hamming space is aso in the code (inear codes) and if the carry is in the code of the next higher eve 134

4 Internationa Zurich Seminar on Information and Communication (IZS), February 21 23, 2018 If in ASK ony the owest eve is coded (identica to construction A [2]), Constraints b) and c) are automaticay fufied and a attice is present However, the (gross) coding gain is then imited to 6 db by the first uncoded eve In QAM the eves are spaced ony by φ = 2 (3 db) and not 2 as in ASK Here, no carry to the next eve but ony the second and third next eve is caused Hence, the ower two eves can be coded and sti a constraints are fufied and a attice is present In this case, the (gross) coding gain is aso imited to 6 db C Mutieve Codes and Integer Linear Combinations We are now interested in (Gaussian) integer inear combinations of mutieve codewords To this end, we consider C from (6), ie, ignore the boundary and the offset Let the MLC codewords c (k) in signa space be obtained via the m codewords c (k), = 0,, m 1, k = 1, 2,, in Hamming space, ie, c (k) = m 1 =0 ψ(c(k) )φ (10) The action of integer inear combinations of these codewords is determined by the binary expansion (wrt the base φ, z (k) i F 2 ) of the respective (Gaussian) integer z k z k = [ z (k) 2 z (k) 1 z (k) 0 ]φ = i 0 ψ(z(k) i ) φ i (11) Using (10) and (11) we have (ψ(a)ψ(b) = ψ(ab), a, b F 2 ) z kc (k) = z m 1 k k k =0 ψ(c(k) ) φ = m 1 =0 i 0 k ψ(z(k) i ) φ i ψ(c (k) ) φ = m 1 =0 i 0 k ψ(z(k) i c (k) ) φ i+ = ψ(ceff ) φ, (12) where is the effective codeword at eve These words can be cacuated by appying (8) or (9) (ASK or QAM) recursivey For exampe, for two codewords and QAM signaing we have 0 = z (1) 0 c(1) 0 z (2) 0 c(2) 0 (13) 1 = z (1) 0 c(1) 1 z (2) 0 c(2) 1 z (1) 1 c(1) 0 z (2) 1 c(2) 0 2 = z (1) 0 c(1) 2 z (2) 0 c(2) 2 z (1) 1 c(1) 1 z (2) 1 c(2) 1 z (1) 2 c(1) 0 z (2) 2 c(2) 0 z (1) 0 c(1) 0 z (2) 0 c(2) 0 Having a ook at the appearing terms and keeping in mind that has to be a vaid codeword of code C the three above demands (7) on the codes are directy cear D Mutistage Decoding Usuay, mutieve codes are decoded via mutistage decoding (MSD), cf Ag 1 Thereby, given r = c + n with c = m 1 =0 ψ(c )φ C, one component code (estimate ĉ ) after the other is decoded, taking into account the decoding resuts of ower eves but ignoring the codes of higher eves (ie, treating them as uncoded) Each decoding step is Ag 1 Mutistage Decoding function ĉ = MSD(r) 1 = 0; r = r // init 2 whie < m { } 3 ĉ = C r // decode eve 4 r +1 = (r ψ(ĉ ))/φ // eiminate known eve 5 = + 1 } m 1 6 ĉ = =0 ψ(ĉ )φ // codeword estimate hence identica to decoding a attice constructed via attice construction A [2] (mutieve code where ony the owest eve is coded) Thus, in a mutistage decoder it is immateria whether the entire code forms (a transate of a subset of) a attice However, it is essentia that the effective word at eve is a vaid codeword from C If integer inear combinations of codewords are to be decoded and Constraint c) of (7) is vioated, carries from ower eves destroy this property and MSD does not work any more E Carry Correction In the foowing, we assume that Constraints a) and b) on the component codes are fufied but Constraint c) is vioated We present a generaization of mutistage decoding which decodes the K inear combinations and eiminates the effect of the carries Let c (1), c (2),, c (K) be the codewords of the K users in signa space (component codewords c (k) 0, c(k) 1,, c(k) m 1, k = 1,, K, in Hamming space) and Z the integer matrix Ignoring the noise for the moment we have r 1 c (1) = Z (14) r K c (K) From the discussion above (cf (12) and (13)), we see that the effective codewords at eve = 0 are obtained from the component codewords at this eves via 1,0 = Z 0 K,0 c (1) 0 c (K) 0 (15) where z (i,j) 0 is the east significant bit (LSB) of z i,j wrt to the basis φ and Z 0 = [z (i,j) 0 ] Having estimates for k,0, k = 1,, K, the origina codewords at eve 0 can hence be obtained by soving (over F 2 ) the set of inear equations (15) Once the codewords at eve 0 are known, the contributions (over C) of the superposition of these eves into the higher eves (carries) can be cacuated via 5 s 1 s K = Q φ {Z ψ(c (1) 0 ) ψ(c (K) 0 ) } (16) 5 Q φ {z} denotes the quantization operation which nus the LSB in the binary expansion wrt φ, ie, with z = [ z 2 z 1 z 0 ] φ we obtain Q φ {z} = [ z 2 z 1 0] φ This nuing is required since ony the contribution to the higher eves but not the current eve is of interest 135

5 Internationa Zurich Seminar on Information and Communication (IZS), February 21 23, 2018 Ag 2 Mutistage Decoding with Carry Correction function [ĉ (1),, ĉ (K) ] = MSD(r) 1 = 0; r k, = r k, k = 1,, K // init 2 whie < m { { } 3 ĉ k, = C rk,, k = 1,, K // decode eve 4 sove (15) and cacuate s k via (16) // cacuate carries 5 r k,+1 = (r k, ψ(ĉ k, ) s k, )/φ // eiminate known 6 = + 1 } interference 7 ĉ (k) m 1 = =0 ψ(ĉ k,)φ // codeword estimates This known interference caused by the codewords at eve 0 can now be eiminated Hence, in a generaized version of MSD, i) the infuence of the effective codeword at eve 0 is eiminated by subtraction of ψ( i,0 ) and ii) the infuence (carries) of the inear combinations of the codewords onto higher eves is eiminated by subtraction of s k This procedure is iterated over the eves In Ag 2, a pseudo-code description of this generaized version of mutistage decoding is given The additiona compexity for the cacuation of the correction terms s k is negigibe compared to the decoding operations F Conditions on Z The fina question is under which conditions does the carry correction work Since the effective codewords are generated via the binary matrix Z 0 (the matrix of LSBs of the (Gaussian) integer entries of the matrix Z), this matrix has to have fu rank 6 In order that Z 0 has fu rank, its determinant (over F 2 ) has to be non-zero We have ψ(det(z 0 )) = mod 2 (det(ψ(z 0 ))) and using the Leibniz formua for the expansion of the determinant, we obtain det(ψ(z 0 )) = det(z) + φg If det(z) G is odd (LSB wrt base φ equa to one) then det(ψ(z 0 )) is aso odd and hence non-zero Unimoduar matrices have an odd determinant If det(z) is even (LSB equa to zero) then det(ψ(z 0 )) is aso even and thus det(z 0 ) = 0 and Z 0 is not invertibe Thus it is proven that for a matrices for which det(z) is odd ( 1 + φg), hence in particuar for unimoduar matrices, carry correction works This imposes restrictions on Z However, it has been shown that using a suited attice reduction agorithm (Minkowski reduction) for iid Gaussian channe matrices even the restriction to unimoduar matrices causes amost no oss compared to the set of fu-rank matrices [12] IV NUMERICAL RESULTS In oder to study the effect of integer inear combinations and carry correction, numerica simuations have been conducted We assume a 16 QAM consteation; the code has four eves First, a toy exampe is presented Linear bock codes of ength N = 13 are empoyed, the generator matrices of the 6 If Z 0 has coumns which are the a-zero vector, the corresponding codewords do not infuence the effective codewords and do not contribute to carry generation Hence, ony the reevant part (matrix Z 0 with a-zero coumns removed) has to have fu rank WER uncoded SISO AWGN usua decoding eve 2 uncoded carry correction og 10 (1/σ2 n ) [db] Fig 3 Word error rate over the inverse noise power (in db) 16 QAM Component codes: inear bock codes of ength N = 13 Rate 31/13 bit/symbo K = 3 users Fixed integer matrix Z = [2 2j 1 2 j; 1 1 j 0; 1 j 0]; det(z) = 2 + j Dotted: asymptotic behavior (13, 2, 8) code at eve 0 and that of the (13, 4, 4) code eve 1 are [ ] G 0 =, G = (17) At eve 2 a (13, 12, 2) singe-parity-check code is used and eve 3 is uncoded ((13, 13, 1) code) This construction does not fufi Condition c) of (7), hence no attice is present and carry correction is required for correct decoding Fig 3 shows the resuts (word error rate averaged over the users) for K = 3 and the arbitrariy seected integer matrix Z = [2 2j 1 2 j; 1 1 j 0; 1 j 0] with det(z) = 2 + j Uncoded transmission and coded transmission over the singe-input/singe-output (SISO) AWGN channe are shown for reference The code has an asymptotic (gross) coding gain of 9 db (dotted; curve for uncoded transmission shifted by the asymptotic gain) If no carry correction is appied, decoding of the mutieve code fais competey 7 With the proposed carry correction the performance of the code over the SISO channe can amost be achieved (some error mutipication is present) If eve 2 is (treated as) uncoded, Constraint c) is fufied and no carry correction is required However, the (asymptotic, gross) coding gain is then imited to 6 db Via carry correction we can break this 6 db barrier Next, a mutieve code with ow-density parity-check (LDPC) codes as component codes is considered The rates of the codes are adjusted according to the capacity design rue for mutieve codes [14] For a target rate of 3 bits per QAM symbo, the rates of the component codes have to be seected as R 0 /R 1 /R 2 /R 3 = 282/753/964/1 We use a code ength N = 5000, which gives code dimensions κ 0 /κ 1 /κ 2 /κ 3 = 1412/3766/4820/ Note that the component decoders aways return a vaid codeword Since, due to the carry, the effective word at eve 2 is not a vaid codeword, wrong correction occurs, eading to a high foor in the error rate 136

6 Internationa Zurich Seminar on Information and Communication (IZS), February 21 23, 2018 BER og 10 (1/σ 2 n) [db] uncoded SISO AWGN usua decoding eve 2 uncoded carry correction Fig 4 Bit error rate over the inverse noise power (in db) 16 QAM Component codes: LDPC codes of ength N = 5000 Rate 3 bit/symbo K = 3 users Fixed integer matrix Z = [1 j 1+j; 1 0 2j; 2j 1 j]; det(z) = j Dotted: asymptotic behavior As LDPC codes irreguar repeat-accumuate codes [8] are used The required subcode property C 0 C 1 C 2 C 3 = F N 2 is guaranteed by constructing the parity-check matrices H in such a way that H +1 is a subset of the rows of H To that end, first the (N κ 2 ) N matrix H 2 is constructed with an utra-sparse eft part (coumn weight 2) and a right staircase part This matrix is extended to H 1 by adding κ 2 κ 1 rows in such a way that the newy added eft part is utra-sparse and that the staircase construction is seamess continued in the right part Given H 1, the fina parity-check matrix H 0 is constructed in the same way by adding κ 1 κ 0 rows Given the parity-check matrices H, generator matrices G for systematic encoding are cacuated Noteworthy, this construction guarantees the subcode property (Constraint b)) of the inear (Constraint a)) component codes However, Constraint c) is neither taken into account nor fufied Fig 4 shows the error rate of the information bits over the noise eve (in db) 8 Here, Z = [1 j 1+j; 1 0 2j; 2j 1 j] with det(z) = j is chosen randomy Messagepassing decoders using og-ikeihood ratios based on nearestneighbor approximation are used; at maximum 10 iterations are performed Basicay, the same behavior as in the exampe above is visibe The exact shape of the fattening in case of conventiona (independent) mutistage decoding depends on the operation of the decoder in case of non-converges (here: the current variabe-node vaues are quantized and output) If eve 2 is treaded as uncoded, the (gross) coding gain is again imited to 6 db Via the proposed generaized decoding scheme, a performance extremey cose to that of the code over the SISO AWGN channe can be achieved V SUMMARY AND CONCLUSIONS In this paper, we have studied the appication of mutieve codes in LRA equaization, ie, the decoding of mutieve 8 Pease note the different scaing of the x-axis compared to Fig 3 codes when (Gaussian) integer inear combinations of the codewords are present A generaized version of mutistage decoding incorporating a carry correction has been proposed which has ony margina additiona compexity compared to independent decoding Thereby, the constraints on the component codes are significanty reaxed and no attice code has to be generated Numerica resuts for arbitrary but fixed integer matrices have been given The next step is the performance evauation over random (iid Gaussian) channe matrices The factorization agorithm in [4] can straightforwardy be generaized such that ony Z matrices with odd determinant are returned However, amost no oss occurs if Minkowski reduction with its restriction to unimoduar matrices is used In summary, via the new decoding procedure, a simpe combination of mutieve codes with ow-compexity LRA equaization is enabed REFERENCES [1] S H Chae, M Jang, SK Ahn, J Park, C Jeong Mutieve Coding Scheme for Integer-Forcing MIMO Receivers With Binary Codes IEEE Trans Wireess Comm, vo 16, no 8, pp , Aug 2017 [2] JH Conway, NJA Soane Sphere Packings, Lattices and Groups Springer Verag, New York, Berin, 3rd edition, 1999 [3] RFH Fischer, C Windpassinger, C Stierstorfer, C Sieg, A Schenk, Ü Abay Lattice-Reduction-Aided MMSE Equaization and the Successive Estimation of Correated Data AEÜ Int Journa of Eectronics and Communications, vo 65, no 8, pp , Aug 2011 [4] RFH Fischer, M Cyran, S Stern Factorization Approaches in Lattice- Reduction-Aided and Integer-Forcing Equaization In 2016 Int Zurich Seminar on Communications, Zurich, Switzerand, March 2016 [5] GD Forney Coset Codes I Introduction and Geometrica Cassification IEEE Trans Information Theory, vo 34, no 5, pp , Sep 1988 [6] WJ Gibert Arithmetic in Compex Bases Mathematics Magazine, vo 57, No 2, pp 77 81, March 1984 [7] H Imai and S Hirakawa A New Mutieve Coding Method Using Error Correcting Codes IEEE Trans Information Theory, vo 23, no 3, pp , May 1977 [8] H Jin, A Khandekar, R McEiece Irreguar Repeat-Accumuate Codes In 2nd Int Symp on Turbo Codes and Re Topics, Brest, France, 2000 [9] W Kositwattanarerk, F Oggier Connections Between Construction D and Reated Constructions of Lattices Designs, Codes and Cryptography, vo 73, no 2, pp , Nov 2014 [10] B Nazer, M Gastpar Compute-and-Forward: Harnessing Interference Through Structured Codes IEEE Trans Information Theory, vo 57, no 10, pp , Oct 2011 [11] W Penney A Binary System for Compex Numbers Journa of the Association for Computing Machinery (JACM), vo 12, no 2, pp , Apri 1965 [12] S Stern, RFH Fischer Optima Factorization in Lattice-Reduction- Aided and Integer-Forcing Linear Equaization In 11 Int ITG Conf on Systems, Communications, and Coding (SCC), Hamburg, Germany, February 2017 [13] G Ungerböck Channe Coding with Mutieve/Phase Signas IEEE Trans Information Theory, vo 28, no 1, pp 55 67, Jan 1982 [14] U Wachsmann, RFH Fischer, JB Huber Mutieve Codes: Theoretica Concepts and Practica Design Rues IEEE Trans Information Theory, vo 45, no 5, pp , Ju 1999 [15] C Windpassinger, RFH Fischer Low-Compexity Near-Maximum- Likeihood Detection and Precoding for MIMO Systems using Lattice Reduction In IEEE Information Theory Workshop, pp , Paris, France, March/Apri 2003 [16] H Yao, G Worne Lattice-Reduction-Aided Detectors for MIMO Communication Systems In IEEE Goba Teecommunications Conference, pp , Taipei, Taiwan, Nov 2002 [17] J Zhan, B Nazer, U Erez, M Gastpar Integer-Forcing Linear Receivers IEEE Trans Information Theory, vo 60, no 12, pp , Dec