Rank Metric Decoder Architectures for Random Linear Network Coding with Error Control

0.39 MB·English

by  Ning Chen

1
Rank Metric Decoder Architectures for Random
Linear Network Coding with Error Control
Ning Chen Member, IEEE, Zhiyuan Yan Senior Member, IEEE, Maximilien Gadouleau Member, IEEE,
Ying Wang Member, IEEE, and Bruce W. Suter Senior Member, IEEE
Abstract—While random linear network coding is a powerful bytraditionalnetworkswithoutnetworkcoding[1].Inaddition
tool for disseminating information in communication networks, to the throughput improvement, network coding is robust to
it is highly susceptible to errors caused by various sources. Due
0 nonergodic network failures and ergodic packet erasures, can
to error propagation, errors greatly deteriorate the throughput
1 be done in a distributed manner with low complexity, and
of network coding and seriously undermine both reliability and
0
security of data. Hence error control for network codingisvital. improvesmobilityandmanagementofwirelessnetworks.Fur-
2
Recently, constant-dimension codes (CDCs), especially Ko¨tter– thermore, implementations of network coding in both wired

ul Kschischang(KK)codes,havebeenproposedforerrorcontrolin and wireless environments demonstrate its practical benefits
randomlinearnetworkcoding.KKcodescanalsobeconstructed
J (see, for example, [3], [4]). Due to these advantages, network
from Gabidulin codes, an important class of rank metric codes.
4 coding is already used or considered in gossip-based data
Rank metric decoders have been recently proposed for both
1 Gabidulin and KK codes, but they have high computational dissemination[5],802.11wirelessadhocnetworking[4],peer-
complexities. Furthermore,itisnot clearwhethersuchdecoders to-peernetworks[3], and mobile ad hocnetworks(MANETs)
]
T are feasible and suitable for hardware implementations. In this [6].
paper, we reduce the complexities of rank metric decoders and
I Random linear network coding (RLNC) [7] is arguablythe
s. prersouplotsseofnoovuerlddeeccooddeerraarrcchhiitteeccttuurreessffoorrbGoatbhidcoudliens.aTnhdeKsyKntchoedseiss most important class of network coding. Instead of using net-
c with limited error-correcting capabilities over small fields show workcodingoperationscentrallydesignedtoachievethemax-
[
that our architectures not only are affordable, but also achieve imumthroughputbasedonthenetworktopology,RLNCtreats

2 high throughput. all packets as vectors over some finite field and forms an
v IndexTerms—Constant-dimensioncodes(CDCs),Decoding,Er- outgoing packet by linearly combining incoming packets us-
1
ror correction coding, Gabidulin codes, Galois fields, Integrated ing random coefficients. Due to its random linear operations,
0
circuits,Ko¨tter–Kschischangcodes,Networkcoding,Rankmetric RLNC not only achieves network capacity in a distributed
6
codes, Subspace codes.
4 manner,butalsoprovidesrobustnesstochangingnetworkcon-
. ditions.Unfortunately,itishighlysusceptibleto errorscaused
9
0 I. INTRODUCTION byvariousreasons,suchasnoise,maliciousormalfunctioning
9 nodes, or insufficient min-cut [8]. Since linearly combining
Network coding [1] is a promising candidate for a new
0 packets results in error propagation, errors greatly deteriorate
unifying design paradigm for communication networks [2].
v: The basic idea of network codingis simple yet powerful.Tra- the throughput of network coding and seriously undermine
i both reliability and security of data. Thus, error control for
X ditionally,anintermediatenodeinanetworksimplystoresand
random linear network coding is critical and has received
forwards each packet it receives. Instead, in network coding,
r
growing attention recently (see, for example, [8], [9]).
a an intermediate node combines several incoming packets into
ErrorcontrolschemesproposedforRLNCassumetwotypes
one or several outgoing packets and forwards. This simple
of transmission models. The schemes of the first type depend
change has a significant ramification: the capacity of a mul-
on and take advantage of the underlying network topology or
ticast (the minimum cut between the source node and any
the particular linear network coding operations performed at
destination node) can be achievedby network coding, but not
various network nodes. The schemes of the second type [8],
This work was supported in part by Thales Communications, Inc. and in [9]assumethatthetransmitterandreceiverhavenoknowledge
part by a summer extension grant from AFRL. The material in this paper of such channeltransfer characteristics. The two transmission
was presented in part at the IEEE Workshop on Signal Processing Systems,
models are referred to as coherent and noncoherent network
Tampere,Finland,October 2009.
NingChenwaswiththeDepartmentofElectricalandComputerEngineer- coding, respectively.
ing, Lehigh University, Bethlehem, PA 18015 USA. Now he is with PMC- Ithasbeenrecentlyshown[8]thatanerrorcontrolcodefor
SierraInc.,Allentown,PA18104USA(e-mail:ning [email protected]).
noncoherent network coding, called a subspace code, is a set
ZhiyuanYaniswiththeDepartmentofElectrical andComputerEngineer-
ing,LehighUniversity,Bethlehem,PA18015USA(e-mail:[email protected]). of subspaces (of a vector space), and information is encoded
MaximilienGadouleauwaswiththeDepartmentofElectricalandComputer in the choice of a subspace as a codeword; a set of packets
Engineering,LehighUniversity,Bethlehem,PA18015USA.Nowheiswith
that generate the chosen subspace is then transmitted [8]. In
CReSTIC, Universite´ de Reims Champagne-Ardenne, Reims 51100 France
(e-mail:[email protected]). contrast, a code in classical coding theory is a set of vectors,
YingWangiswithQualcommFlarionTechnologies,Bridgewater,NJ08807 andinformationisencodedinthechoiceofavectorasacode-
USA(e-mail: [email protected]).
word. A subspace code is called a constant-dimension code
Bruce W.Suter is with AirForce Research Laboratory, Rome, New York
13441USA(e-mail: [email protected]). (CDCs) if its subspaces are of the same dimension. CDCs
2
are of particular interest since they lead to simplified network researchresultsleadtoefficientdecodersandhencereducethe
protocols due to the fixed dimension. A class of asymptoti- area and power overheads of network coding.
cally optimal CDCs have been proposed in [8], and they are In thispaper,we focuson the generalizedGabidulindecod-
referred to as the KK codes. A decoding algorithm based ing algorithm [9] for the KK codes and decoding algorithm
on interpolation for bivariate linearized polynomials is also in [16] for Gabidulin codes for two reasons. First, compared
proposedin[8]fortheKKcodes.ItwasshownthatKKcodes with the decoding algorithm in [8], the generalized Gabidulin
correspondto lifting [9] of Gabidulincodes[10], [11], a class decoding [9] has a smaller complexity, especially for high-
of optimal rank metric codes. Gabidulin codes are also called rate KK codes [9]. Second, components in the errors-only
maximum rank distance (MRD) codes, since they achieve the Gabidulindecodingalgorithmin[16]canbeeasilyadaptedin
Singleton bound in the rank metric [10], as RS codes achieve thegeneralizedGabidulindecodingofKKcodes.Thus,among
the Singleton bound of Hamming distance. Gabidulin codes the decoding algorithms for Gabidulin codes, we focus on
have found applications in storage systems [11], space-time the decodingalgorithmin [16]. Finally, althoughwe focuson
coding[12]aswellascryptography[13],andvariousdecoding randomlinear networkcodingwith errorcontrolin thispaper,
algorithmshavebeen proposedforGabidulincodes[14]–[18]. our results can be easily applied to random linear network
Due to the connection between Gabidulin and KK codes, the coding without error control. For random linear network cod-
decodingofKKcodescanbeviewedasgeneralizeddecoding ingwithouterrorcontrol,thedecodingcomplexityisprimarily
ofGabidulincodes,whichinvolvesdeviationsaswellaserrors due to invertingof the globalcoding matrix via Gauss-Jordan
anderasures[9].Gabidulincodesaresignificantinthemselves. elimination, which is considered in this paper.
For coherent network coding, the error correction capability One of our main contributions include several algorithmic
of error control schemes is succinctly described by the rank reformulations that reduce the computational complexities of
metric [19]; thus error control codes for coherent network decoders for both Gabidulin and KK codes. Our complexity-
coding are essentially rank metric codes. saving algorithmic reformulations are:
The benefits of network coding above come at the price of • We first adopt normal basis representations for all finite
additionaloperationsneededatthesourcenodesforencoding, fieldelements,andthensignificantlyreducethecomplex-
at the intermediate nodes for linear combining, and at the ity of bit-parallel normal basis multipliers by using our
destination node(s) for decoding. In practice, the decoding common subexpression elimination (CSE) algorithm;
complexities at destination nodes are much greater than the • ThedecodingalgorithmsofbothGabidulinandKKcodes
encodingandcombiningcomplexities.Thedecodingcomplex- involve solving key equations. Based on approaches in
ities of RLNC are particularly high when large underlying [23], [24], we reformulate the Berlekamp–Massey algo-
fields are assumed and when additional mechanisms such as rithm (BMA), and propose an inversionless BMA. Our
error control are accounted for. Clearly, the decoding com- inversionless BMA does not require any finite field in-
plexities of RLNC are critical to both software and hardware versions, and leads to reduced complexities as well as
implementations. Furthermore, area/power overheads of their efficient architectures;
VLSI implementationsare importantfactors in system design. • The decoding algorithm of KK codes requires that the
Unfortunately, prior research efforts have mostly focused on input be arranged in a row reduced echelon (RRE) form.
theoretical aspects of network coding, and complexity reduc- We define a more generalized form called n-RRE form,
tion and efficient VLSI implementation of network coding and show that it is sufficient if the inputis in the n-RRE
decoders have not been sufficiently investigated so far. For form.Thischangenotonlyreducesthe complexityof re-
example,althoughthedecodingcomplexitiesofGabidulinand formulatingtheinput,butalsoenablesparallelprocessing
KKcodeswereanalyzedin[20],[21]and[8],[9],respectively, of decoding KK codes based on Cartesian products.
buttheydonotreflecttheimpactofthesizeoftheunderlying Theothermaincontributionofthispaperisefficientdecoder
finite fields. To ensure high probability of success for RLNC, architectures for both Gabidulin and KK codes. Aiming to
a field of size 28 or 216 is desired [22]. However, these large reduce the area and to improve the regularity of our decoder
fieldsizeswillincreasedecodingcomplexitiesandhencecom- architectures, we have also reformulated other steps in the
plicate hardware implementations. Finally, to the best of our decoding algorithm. To evaluate the performance of our de-
knowledge,hardwarearchitecturesforthesedecodershavenot coderarchitecturesforGabidulinandKKcodes,weimplement
been investigated in the open literature. our decoderarchitecturefor two rate-1/2Gabidulin codes, an
In this paper, we fill this significant gap by investigating (8,4) code and a (16,8) code, and their corresponding KK
complexityreductionsandefficienthardwareimplementations codes. Our KK decoders can be used in network coding with
fordecodersinRLNCwitherrorcontrol.Thiseffortissignifi- variouspacketlengthsbyCartesianproduct[9].Thesynthesis
canttotheevaluationanddesignofnetworkcodingforseveral results of our decoders show that our decoder architectures
reasons.First,ourresultsevaluatethecomplexitiesofdecoders for Gabidulin and KK codes over small fields with limited
for RLNC as well as the area, power, and throughputof their error-correcting capabilities not only are affordable, but also
hardware implementations, thereby helping to determine the achievehighthroughput.Ourdecoderarchitecturesandimple-
feasibility and suitability of network coding for various ap- mentation results are novel to the best of our knowledge.
plications. Second, our research results provide instrumental Therestofthepaperisorganizedasfollows.Afterbrieflyre-
guidelines to the design of network coding from the perspec- viewingthebackgroundinSectionII,wepresentourcomplexity-
tiveofcomplexityaswellhardwareimplementation.Third,our saving algorithmic reformulations and efficient decoder archi-
3
tectures in Sections III and IV, respectively. In Section V, B. Gabidulin Codes and Their Decoding
the proposed architectures are implemented in Verilog and A Gabidulin code [10] is a linear (n,k) code over F ,
qm
synthesized for area/performance evaluation. The conclusion
whose parity-check matrix has a form as
is given in Section VI.
h[0] h[0] ··· h[0]
0 1 n−1
h[1] h[1] ··· h[1]
 0 1 n−1 
H = ... ... ... ...  (1)
II. PRELIMINARIES h[n−k−1] h[n−k−1] ··· h[n−k−1]
 0 1 n−1 
where h0,h1,...,hn−1 ∈ Fqm are linearly independent over
A. Notation Fq. SinceFqm isanm-dimensionalvectorspaceoverFq, itis
necessary that n ≤ m. Conversely, a generator matrix of this
Let q denote a power of prime and F denote a finite Gabidulin code is given by
qm
fimealtdriocefsoordveerrqFmq.aWndeuusseeIFnqnt×omdetnoodteenaontent×hensiedteonftiatyllmna×trmix. gg0[[10]] gg1[[10]] ...... ggn[[10−]]1
sFuobrsaets{e0t,U1,⊆...{,0n,1−,.1.}.\,nU−an1d}I,UUdcedneontoestetshethceocluommnpsleomfeInnt G= 0... 1... ... n...−1 (2)
in U. In this paper, all vectors and matrices are in bold face. g[k−1] g[k−1] ... g[k−1]
 0 1 n−1 
The rank weight of a vector over F is defined as the  
qm where n−1g[k−1]h[s] =0,fors=0,1,...,n−1. The mini-
maximal number of its coordinates that are linearly indepen- i=0 i i
dentoverthebasefieldF .Rankmetricistheweightofvector mum rank distance of a Gabidulin code is d=n−k+1 and
q P
difference[25].ForacolumnvectorX ∈Fn ,wecanexpand all Gabidulin codes are MRD codes.
qm
The decoding process of Gabidulin codes includes five ma-
each of its component into a row vector over the base field
F . Such a row expansion leads to an n×m matrix over F . jor steps: syndrome computation,key equationsolver, finding
q q
the root space, finding the error locators by Gabidulin’s al-
In this paper, we slightly abuse the notation so that X can
represent a vector in Fn or a matrix in Fn×m, although the gorithm [10], and finding error locations. The data flow of
qm q
Gabidulin decoding is shown in Figure 1.
meaning is usually clear given the context.
MostkeyequationsolversarebasedonamodifiedBerlekamp–
Given a matrix X, its row space, rank, and reduced row
Masseyalgorithm(BMA)[16]oramodifiedWelch–Berlekamp
echelon (RRE) form [26] are denoted by hXi, rankX, and
algorithm(WBA)[29].Otherdecodingalgorithmsinclude[30],
RRE(X), respectively. For a subspace hXi, its dimension is
[31]. The decoding algorithms in [31] are for cryptography
denoted by dimhXi and rankX = dimhXi. The rank dis-
applications and have very high complexity. In this paper, we
tanceoftwovectorsX andY inFn isdefinedasd (X,Y),
qm R focus on the modified BMA.
rank(X−Y). The subspace distance [8] of their row spaces
AsinRSdecoding,wecancomputesyndromesforGabidu-
2hXdiim,h(YhXiiis∩dheYfinie)d. asdS(hXi,hYi),dimhXi+dimhYi− vlienctcoordre.sTahseSnt=he(sSy0n,dSro1m,.e..p,oSlydn−o2m) ,ialHS(rx)fo=r anyd−r2ecSeixve[jd]
ˆ ˆ j=0 j
Let the received matrix be Y = [A | y], where A ∈ can be used to solve the key equation
FN×n and y ∈ FN×m. Note that we always assume the P
reqceived matrix is fqull-rank [9]. The row and column rank σ(x)⊗S(x)≡ω(x)modx[d−1] (3)
ˆ ˆ ˆ
deficiencies of A are δ =N −rankA and µ=n−rankA,
for the error span polynomial σ(x), using the BMA. Up to
respectively.Then the RRE form of Y is expandedinto Y¯ =
t=⌊(d−1)/2⌋ error values E ’s can be obtained by finding
[IU0c I0δ]RRE(Y) = [In+0LˆIUT Erˆ], where Uc denotes the col- a basis E0,E1,... for the rootjspace of σ(x) using the meth-
umnpositionsofleadingentriesinthefirstnrowsofRRE(Y). ods in [32], [33]. Then we can find the error locators X ’s
ˆ ˆ j
The tuple (r,L,E) is called a reduction of Y [9]. corresponding to E ’s by solving a system of equations
j
Polynomials over F with only non-zero terms with de-
qm τ−1
grees qi are called linearized polynomials[27], [28]. For con- S = X[l]E , l=0,1,...,d−2 (4)
venience, let [i] denote qi. In a linearized polynomial, the l j j
j=0
X
greatestvalueofiofnon-zerotermsisdefinedasitsq-degree.
where τ is the number of errors. Gabidulin’s algorithm [10]
The symbolicproductoftwo linearizedpolynomialsa(x) and
in Algorithm 1 can be used to solve (4). Finally, the error
b(x), denoted by ⊗ (that is, a(x)⊗b(x) = a(b(x))), is also
locations L ’s are obtained from X ’s by solving
a linearized polynomial.The q-reverse of a linearized polyno- j j
mial f(x) = p f x[i] is given by the polynomial g(x) = n−1
i=0 i
pi=0gix[i], wPhere gi = fp[i−−ip] for i = 0,1,...,p and p Xj = Lj,khi, j =0,1,...,τ −1. (5)
is the q-degree of f(x). For a set α of field elements, we Xi=0
P
use minpoly(α) to denote its minimal linearized polynomial, Algorithm 1 (Gabidulin’s Algorithm [10]).
whichis the moniclinearizedpolynomialof least degreesuch Input: S0,S1,...,Sd−2 and E0,E1,...,Eτ−1
that all the elements of α are its roots. Output: X0,X1,...,Xτ−1
4
r S σ(x)
Received Syndromes BMA
X E
Corrected Error Gabidulin’s Roots
Fig.1. DataflowofGabidulin decoding
1.1 Compute τ ×τ matrices A and Q as n are needed, which results in prohibitively high complexity
duetothehugefieldsize ofF .A low-complexityapproach
qm
E i=0
j in [9] suggested that instead of using a single long Gabidulin
Ai,j =0, i6=0,j <i code,a CartesianproductofmanyshortGabidulincodeswith
Ai−1,j −Ai−1,i−1(AAi−i−1,1i,−j1)[−1] i6=0,j ≥i thesamedistancecanbeusedtoconstructconstant-dimension
codes for long packets via the lifting operation.
Q =Sj i=0 Inthedecodingalgorithmof[9],thematrixY isfirstturned
i,j (Qi−1,j −Ai−1,i−1(QAii−−11,,ji−+11)[−1] otherwise. into an RˆREˆform. Based on an RRE form of Y, a three-
tuple (r,L,E), referred to as a reduction of Y, is obtained.
1.2 CanodmXpuite=X(iQ’sir,0ec−ursivjτe=−lyi1+a1sAXiτ,j−X1j=)/AQiτ,i−,1f,0o/rAiτ=−1,ττ−−1 Itwasproved[9]thaˆtdS(hXi,hYi)=2ˆrank[L0ˆ r−Eˆx]−µ−δ,
2,τ −3,...,0. whereµ=n−rankLandδ =N−rankL.Nowthedecoding
P problemtominimizethesubspacedistancebecomesaproblem
In total, the decoding complexity of Gabidulin codes is to minimize the rank distance.
roughly O(n2(1 − R)) operations over Fqm [20], where R The generalized rank decoding [9] finds an error word eˆ=
is thecoderate or O(dm3)operationsoverFq [21]. Note that argmine∈r−Crank[Lˆ0 Eeˆ]. The error word eˆis expanded as a
allpolynomialsinvolvedinthedecodingprocessarelinearized summation of products of column and row vectors [9] such
polynomials. that eˆ = τ−1L E . Each term L E is called either an
j=0 j j j j
Gabidulincodesareoftenviewedasthecounterpartinrank erasure, if L is known, or a deviation, if E is known, or
j j
metric codes of the well-known Reed–Solomon codes. As an error, iPf neither L nor E is known. In this general de-
j j
shown in Table I, an analogy between Reed–Solomon and coding problem, L has µ columns from Lˆ and E has δ rows
Gabidulin codes can be established in many aspects. Such an from Eˆ. Given a Gabidulin code of minimum distance d, the
analogyhelpsus understandthe decodingof Gabidulin codes, correspondingKK code is able to correct ǫ errors, µ erasures,
andinsomecasesallowsustoadaptinnovationsproposedfor and δ deviations as long as if 2ǫ+µ+δ <d.
Reed–Solomon codes to Gabidulin code. Algorithm 2 was proposed [9] for generalized decoding of
the KK codes, and its data flow is shown in Figure 2. It
TABLEI
ANALOGYBETWEENREED–SOLOMONANDGABIDULINCODES requires O(dm) operations in Fqm [9].
Reed–Solomon Gabidulin Algorithm 2 (General Rank Decoding [9]).
ˆ ˆ
Metric Hamming Rank Input: received tuple (r,L,E)
RDienggreoef Polyniomials Lineariz[eid]=Poqlyinomials Output: error word eˆ
KeyOperation PolynomialMultiplication SymbolicProduct 2.1 Compute S =Hr, Xˆ =LˆTh, λ (x)=minpoly(Xˆ),
GenerationMatrix [gi] [g[i]] ˆ U
j j σ (x)=minpoly(E), andS (x)=σ (x)⊗S(x)⊗
ParityCheckMatrix [hi] [h[i]] D DU D
j j ζ (x), where ζ (x) is the q-reverse of λ (x).
KeyEquationSolver BMA ModifiedBMA U U U
ErrorLocations Roots RootSpaceBasis 2.2 Compute the error span polynomial:
ErrorValueSolver Forney’sFormula Gabidulin’sAlgorithm
a) Use the modifiedBMA [16]to solve thekeyequa-
tion σ (x) ⊗ S (x) ≡ ω(x)modx[d−1] such
F DU
that degω(x)<[τ] where τ =ǫ+µ+δ.
C. KK Codes and Their Decoding b) Compute SFD(x)=σF(x)⊗σD(x)⊗S(x).
c) UseGabidulin’salgorithm[10]tofindβthatsolves
By the lifting operation [9], KK codes can be constructed
S = µ−1X[l]β ,l = d−2,d−3,...,d−
from Gabidulin codes. Lifting can also be seen as a gener- FD,l j=0 j j
1−µ.
alization of the standard approach to random linear network
P
d) Computeσ (x)=minpoly(β)followedbyσ(x)=
coding[7], which transmits matricesin the formX =[I |x], U
where X ∈ Fn×M, x ∈ Fn×m, and m = M −n. Hence by σU(x)⊗σF(x)⊗σD(x).
q q
addingtheconstraintthatxistherowexpansionofcodewords 2.3 Find a basis E for the root space of σ(x).
from a Gabidulin code C over F , error control is enabled 2.4 Find the error locations:
qm
in network coding. a) SolveS = τ−1X[l]E ,l=0,1,...,d−2using
l j=0 j j
Inpractice,thepacketlengthcouldbeverylong.Toaccom- Gabidulin’salgorithm[10]tofindtheerrorlocators
modatelongpacketsbasedontheKKcodes,verylargemand X0,X1,...,PXτ−1 ∈Fqm.
5
b) Compute the error locations Lj’s by solving (5). becomes part of a cyclic convolution of (h0,h1,...,hm−1)
c) Compute the error word eˆ= τ L E , where andr,forwhichfastalgorithmsareavailable[35],[36].Using
j=1 j j
each E is the row expansion of E . fast cyclic convolution algorithms are favorable when m is
j j
P
large.
III. COMPUTATIONALCOMPLEXITY REDUCTION
In general, RLNC is carried out over F , where q is any B. Normal Basis Arithmetic Operations
q
prime power. That is, packets are treated as vectors over F . We also propose finite field arithmetic operations with re-
q
Since our investigation of computational complexities is for ducedcomplexities, when normalbasis representationis used.
bothsoftwareandhardwareimplementationsofRLNC,where When represented by vectors, the addition and subtraction of
dataarestoredandtransmittedinbits,wefocusonRLNCover two elements are simply component-wise addition, which is
characteristic-2 fields in our work, i.e., q is a power of two. straightforward to implement. For characteristic-2 fields F2m,
In some cases, we further assume q =2, as it leads to further inverses can be obtained efficiently by a sequence of squar-
complexity reductions. ing and multiplying, since β−1 = β2m−2 = β2β4...β2m−1
for β ∈ F [34]. Since computing [i]th powers requires
2m
no computation, the complexity of inversion in turn relies
A. Finite Field Representation
on that of multiplication. Division can be implemented by a
Finite field elements can be represented by vectors using concatenationof inversionand multiplication:α/β =α·β−1,
different types of bases: polynomial basis, normal basis, and
and hence the complexity of division also depends on that of
dual basis [34]. In rank metric decoders, most polynomials
multiplication in the end.
involved are linearized polynomials, and hence their evalu-
There are serial and parallel architectures for normal basis
ations and symbolic products require computing their [i]th
finite field multipliers. To achieve high throughput in our de-
powers. Suppose a field element is represented by a vector
coder, we consider only parallel architectures. Most normal
overF withrespecttoanormalbasis,computing[i]thpowers
p basis multipliers are based on the Massey–Omura (MO) ar-
(i is a positive or negative integer) of the element is simply
chitecture [34], [37]. The complexity of a serial MO normal
cyclicshifts ofthe correspondingvectorbyi positions,which
basismultiplier,C ,isdefinedasthenumberoftermsa b in
N i i
significantlyreducescomputationalcomplexities.Forexample,
computingabitofthe productc=ab,wherea ’sandb ’s are
i i
the computationalcomplexityofAlgorithm1 is primarilydue
the bits of a and b, respectively. It has been shown [38] that
to the following updates in Step 1.1: a parallel MO multiplier over F2mneeds m2 AND gates and
Ai,j =Ai−1,j −(AAi−i−1,1i,−j1)[−1]Ai−1,i−1 (6) rfiaeteslmpdesocsFttiv2m8el(yaCn[dN34F+].21mU6,s−itnhge2i)ar/c2CoNXm’OmsRoarngesamutebisen.xiFmporierzseisdniosttnoane2cl1iem,aifnnodartit8oh5ne,
Qi,j =Qi−1,j −(Qi−1,j+1)[−1]Ai−1,i−1 algorithm [39], we significantly reduce the number of XOR
Ai−1,i−1
gates while maintaining the same critical path delays (CPDs)
which require divisions and computing [−1]th powers. With of one AND plus five XOR gates and one AND plus seven
normal basis representation, [−1]th powers are obtained by XORgatesasdirectimplementations,respectively.Ourresults
a single cyclic shift. Furthermore, when q = 2, they can be are comparedto those in [34], [38] in Table II, wherewe also
computed in an inversionless form provide the prime polynomial P(x) for each field.
Ai,j =Ai−1,j −(Ai−1,jAi−1,i−1)[−1] (7) parTtihceulraerdlyucimedpograttaentcfoournhtafrodrwnaorermimalplbeamsiesntmatuioltnipsloicfaRtiLonNCis.
Qi,j =Qi−1,j −(Qi−1,j+1Ai−1,i−1)[−1], Thisimprovementistransparenttothecomplexityofdecoders,
in terms of finite field operations. When decoders for RLNC
which also avoids finite field divisions or inversions. Thus
are realized in hardware, the reduced gate count for normal
using normalbasis representationalso reducesthe complexity
basismultiplicationwillbereflectedinreducedareaandpower
of Gabidulin’s algorithm.
consumption.
In addition to reduce complexities of finite field arithmetic
operations, normal basis representation leads to reduced com-
plexities in the decoding of Gabidulin and KK codes for sev- C. Inversionless BMA
eral reasons. First, it was shown that using normal basis can
The modifiedBMA for rankmetric codes [16] is similar to
facilitate the computation of symbolic product [20]. Second,
theBMA forRS codesexceptthatpolynomialmultiplications
it was also suggested[20] that solving(5) can be trivialusing
are replaced by symbolic products. The modified BMA [16]
normal basis. When a normal basis is used as hi’s and also requires finite field divisions, which are more complex than
used for representation, the matrix h, whose rows are vector
other arithmetic operations. Following the idea of inversion-
representationsof h ’s with respectto the basis h ’s, becomes
i i less RS decoder [23], we propose an inversionless variant in
an identity matrix with additional all-zero columns. Hence
Algorithm 3.
solving (5) requires no computation. These two complexity
reductionswerealsoobservedin[21].Third,ifanormalbasis Algorithm 3. iBMA
of F2m is usedas hi’s andn=m, the paritycheckmatrixH Input: Syndromes S
in (1) becomes a cyclic matrix. Thus syndrome computation Output: Λ(x)
6
Xˆ λU(x)
×h MinPoly
ˆ
L Eˆ σD(x) SDU(x) σF(x) SFD(x)
Received RRE MinPoly SymProd BMA SymProd
rˆ S
Syndromes
X E σ(x) σU(x) β
Corrected Error Gabidulin’s Roots SymProd MinPoly Gabidulin’s
Fig.2. DataflowofKKdecoding
TABLEII
COMPLEXITIESOFBIT-PARALLELNORMALBASISMULTIPLIERSOVERFINITEFIELDS(FORTHESETWOFIELDS,ALLTHREEIMPLEMENTATIONSHAVE
THESAMECPD.)
XOR
AND
P(x) direct [34] [38] Ours
F2m - m2 m(CN −1) m(CN +m−2)/2 -
F28 (P8i=0xi)−x6−x4−x2 64 160 108 88
F216 (P1i=60xi)−x14−x9−x6−x4 256 1344 792 491
3.1 Initialize: Λ(0)(x) = B(0)(x) = x[0], Γ(0) = 1, and veryhigh;Second,it cannotfind a set of linearlyindependent
L=0. roots.
3.2 For r =0,1,...,2t−1, A probabilistic algorithm to find the root space was pro-
a) Compute the discrepancy ∆ = L Λ(r)S[j] . posed in [33]. For Gabidulin codes, it can be further sim-
r j=0 j r−j
plified as suggested in [20]. But hardware implementations
b) If ∆ =0, then go to (e).
r
c) Modify the connection polynomPial: Λ(r+1)(x) = of probabilistic algorithmsrequire randomnumbergenerators.
(Γ(r))[1]Λ(r)(x)−∆ x[1]⊗B(r)(x). Furthermore,thealgorithmin[33]requiressymboliclongdivi-
r
sion, which is also not suitable for hardwareimplementations.
d) If 2L > r, go to (e). Otherwise, L = r+1−L,
Γ(r+1) =∆ , and B(r)(x)=Λ(r)(x). Go to (a). According to [9], the average complexity of the probabilistic
e) Set Γ(r+1) r= (Γ(r))[1] and B(r+1)(x) = x[1] ⊗ algorithmin[33]isO(dm)operationsoverFqm,whilethatof
B(r)(x). Berlekamp’s deterministic method [32] is O(dm) operations
in F plus O(m3) operations in F . Thus their complexity
3.3 Set Λ(x)=Λ(2t)(x). qm q
difference is small, and hence we focus on the deterministic
method since it is much easier to implement.
Using a similar approach as in [23], we prove that the
Supposeweneedtofindtherootspaceofalinearizedpoly-
output Λ(x) of Algorithm 3 is the same as σ(x) produced
by the modified BMA, except it is scaled by a constant C = nomialr(x), Berlekamp’sdeterministicmethodfirstevaluates
t−1(Γ(2i))[1]. However,this scaling is inconsequentialsince thepolynomialr(x)onabasisofthefield(α0,α1,...,αm−1)
i=0
suchthatv =r(α ),i=0,1,...,m−1.Thenitexpandsv ’s
the two polynomials have the same root space. i i i
Q in the base field as columnsof an m×m matrix V and finds
Using normal basis, the modified BMA in [16] requires at
linearly independent roots z such that Vz = 0. Using the
most ⌊(d−2)/2⌋ inversions, (d−1)(d−2) multiplications,
and (d−1)(d−2) additionsoverF [20]. Our inversionless representation based on (α0,α1,...,αm−1), the roots z are
qm
also the roots of the given polynomial. Finding z is to obtain
version, Algorithm 3, requires at most (3/2)d(d−1) multi-
thelineardependentcombinationsofthecolumnsofV,which
plications and (d−1)(d−2) additions. Since a normal basis
can be done by Gaussian elimination.
inversion is obtained by m−1 normal basis multiplications,
the complexity of normal basis inversion is roughly m − 1
times that of normalbasis multiplication.Hence, Algorithm 3 E. n-RRE Form
reducesthecomplexityconsiderably.Algorithm3isalsomore Given a received subspace spanned by a set of received
suitableforhardwareimplementation,asshowninSectionIV. packets, the input of Algorithm 2 is a three-tuple, called a
reduction of the received space represented by its generator
matrix Y; the three-tuple is obtained based on Y when it is
D. Finding the Root Space
in its RRE form [9]. Thus, before the decoding starts, prepro-
Insteadof findingroots ofpolynomialsin RS decoding,we cessing is performed on the received packets so as to obtain
need to find the root spaces of linearized polynomialsin rank the RRE form of Y. We show that Y needs to satisfy only a
metric decoding.Hencethe Chien searchin RS decodingwill relaxedconstraint.TherelaxedconstraintonY doesnotaffect
haveahighcomplexityfortworeasons.First,itrequirespoly- the decodingoutcome,while leading to two advantages.First,
nomial evaluations over the whole field, whose complexity is the relaxed constraint results in reduced complexities in the
7
preprocessing step. Second and more importantly, the relaxed matrix Y, the preprocessing needed to transform Y into its
constraint enables parallel processing of decoding KK codes n-RRE form is onlypart of the preprocessingto transformY
based on Cartesian products. into its RRE form. We can show that the maximal number
Wefirstdefineann-RREformforreceivedmatrices.Given of arithmetic operations in the former preprocessing is given
ma matraitxrixYYis=in[Aiˆts|ny-]R, wREherfeorAmˆ∈asFlNqon×gnaasnAdˆy(i∈tsFleNqft×mmo,stthne tbeyr (pNrep−ro1ce)ssinir=agn0kisAˆ(−N1(n−+1)m−ranik)(,Yw)−h1e(rnea+s tmhat−oif).thSeinlcaet-
columns) is in its RRE form. Compared with the RRE form, rankY ≥ rPankAˆ, the relaxedi=c0onstraint leads to a lower
the n-RRE form is more relaxed as it puts no constraints on complexity,andthereductionPdependsonrankY andrankAˆ.
the rightpart. We note that an n-RRE form of a matrix is not Second,thereductionforn-RRE formsiscompletelydeter-
unique. mined by the n leftmost columns of Y instead of the whole
We first showthat therelaxedconstraintdoesnotaffectthe matrix,whichgreatlysimplifieshardwareimplementations.This
decoding. Similar to [40, Proposition 7], we first show that a advantageisparticularlyimportantforthedecodingofconstant-
reduction based on n-RRE form of Y always exists. Given dimension codes that are lifted from Cartesian products of
ˆ ˆ ˆ
Y = [A | y] and RRE(A) = RA where the matrix R Gabidulin codes. First, since the row operations to obtain an
represents the reducing operations, the product Y¯′ = RY = n-RRE form depend on Aˆ only, decoding [Aˆ|y |y |···|
0 1
[B′ | Z′] is in its n-RRE form. We note that B′ ∈ FNq ×n yl−1] can be divided into parallel and smaller decoding prob-
and Z ∈ FNq ×m, where the column and row rank deficiency lemswhoseinputsare[Aˆ|y0],[Aˆ|y1],...,[Aˆ|yl−1].Thus,
of B′ are given by µ′ =n−rankB′ and δ′ =N −rankB′, for these constant-dimensioncodes, we can decodein a serial
respectively.We have the followingresult aboutthe reduction manner with only one small decoder, or in a partly parallel
based on Y¯′. fashionwithmoredecoders,oreveninafullyparallelfashion.
Lemma 1. Let Y¯′ and µ′ and δ′ be defined as above. There This flexibility allows tradeoffs between cost/area/power and
ˆ
existsa tuple(r′,Lˆ′,Eˆ′)∈Fn×m×Fn×µ′×Fδ′×m anda set throughput. Furthermore, since the erasures L is determined
U′ satisfying q q q byAˆandisthesamethroughoutall[A|yi],thecomputation
ˆ
of X and λ (x) in Algorithm 2 can be shared among these
U
′ ′
|U | = µ paralleldecodingproblems,therebyreducingoverallcomplex-
IT r′ = 0 ity.
U′
IUT′Lˆ′ = −Iµ′×µ′
rankEˆ′ = δ′ F. Finding Minimal Linearized Polynomials
Minimallinearizedpolynomialscanbecomputedbysolving
so that
In+L0ˆ′IUT′ Erˆ′′ =hY¯′i=hYi. (8) smyisnteimmaslolifnleinareiazredeqpuoaltyionnosm.iGalivxe[np]r+ootspiβ=−00,1βa1i,x.[i.].s,aβtpis−fi1e,sthe
Dh iE
β[0] β[1] ··· β[p−1] aP β[p]
0 0 0 0 0
TbahsLeeedkPmeroymonooaffn:1o-SRuserRheoaEAlwtefpsorpnrtemhantaidtvoiewxfreYAed.c,uaicnntisfiotennadodfaonYfaailnstetRrhnaRattEitvhefeorrreemdduuoccfttiiYoonn. ββp[10[...−0]]1 ββp[11[...−1]]1 ··.··.··. ββ1p[[pp−...−−111]]aap...−11=ββp[1p[...−p]]1. (9)
ismostlydeterminedbythefirstncolumnsofRRE(Y).Also,     
Thus it can be solved by Gaussian elimination over the exten-
this alternative reduction does not come as a surprise. As
shownin[40,Proposition8],rowoperationsonEˆcanproduce sionfieldFqm.Gabidulin’salgorithmisnotapplicablebecause
therowsofthematrixarenotthepowersofthesameelement.
alternative reductions. Next, we show that decoding based on
The complexity to solve (9) is very high. Instead, we re-
ouralternativereductionisthesameasin[40].Similarto[40,
formulate the method from [27, Theorem 7]. The main idea
Theorem 9], we have the following results.
of [27, Theorem 7] is to recursively construct the minimal
Lemma 2. Let (r′,Lˆ′,Eˆ′) be a reduction of Y determined linearizedpolynomialusingsymbolicproductsinsteadofpoly-
by its n-RRE form, we have nomial multiplications in polynomial interpolation. Given lin-
dS(hXi,hYi)=2rank Lˆ0′ r′Eˆ−′x −µ′−δ′. esearrileysinodfelpineenadreiznetdropootlsywno0m,wia1l,s.a.s.,wp−1, we can constructa
(cid:20) (cid:21)
F (x)=x[1]−w x[0]
0 0
Proof: See Appendix B. F1(x)=(x[1]−(F0(w1))q−1x[0])⊗F0(x)
Lemma 2 shows that the subspace decoding problem is ...
ethqeuiavlatelernntattiovethreegdeuncetiroanliz(erd′,GLˆa′b,iEdˆu′)li,nwdheiccohdiinsgopbrtoaibnleedmfwroitmh Fp−1(x)=(x[1]−(Fp−2(wp−1))q−1x[0])⊗Fp−2(x).
an n-RRE formof Y. We illustrate an exampleof KK decod- In the methodin [27, Theorem7], the evaluationof F (w )
i j
ing based on n-RRE approach in Appendix C. has an increasing complexity when the degree of F (x) gets
i
Our alternative reduction leads to two advantages. First, higher.Tofacilitatetheimplementation,wereformulatetheal-
it results in reduced complexity in preprocessing. Given a gorithmtodividetheevaluationintomultiplesteps.Theideais
8
basedonthefactthatFi(wi+1)=(x[1]−(Fi−1(wi))q−1x[0])⊗ Similarly, B(x) and Θ(x) can be combined into one polyno-
Fi−1(wi+1). The reformulated algorithm is given in Algo- mial Θ˜(x). These changes are incorporated in our RiBMA
rithm 4. Thus the evaluation can also be done in a recursive algorithm, shown in Algorithm 5.
way.
Algorithm 5. RiBMA
Algorithm4(AlgorithmforMinimalLinearizedPolynomials). Input: Syndromes S
Output: Λ(x)
Input: Roots w0,w1,...,wp−1 5.1 Initialize:∆˜(0)(x)=Θ˜(0)(x)= 2t−1S x[i],Γ(0) =1,
Output: The minimal linearized polynomial F(p)(x) ∆˜(0) =Θ˜(0) =1, and b=0. i=0 i
3t 3t P
4.1 Initializeγ(0) =w ,forj =0,1,...,p−1andF(0)(x)= 5.2 For r =0,1,...,2t−1,
j j
x[0]. a) Modify the combined polynomial: ∆˜(r+1)(x) =
4.2 For i=0,1,...,p−1, if γ(i) 6=0, Γ(r)∆˜(r)(x)−∆˜(r)Θ˜(r)(x);
i 0
a) F(i+1)(x)=(F(i)(x))[1]+(γ(i))q−1F(i)(x); b) Set b=b+1;
b) For j =i+1,i+2,...,p−1,iγ(i+1) =(γ(i))[1]+ c) If ∆˜0(r) 6=0 andb>0,set b=−b,Γ(r+1) =∆˜(0r),
(γ(i))q−1γ(i). j j and Θ˜(r)(x)=∆˜(r)(x);
i j d) Set ∆˜(r+1)(x) = 3t−1∆˜(r+1)x[i], Θ˜(r)(x) =
i=0 i+1
Sincepowersofqrequireonlycyclicshifting,theoperations 3t−1Θ˜(r) x[i];
i=0 i+1 P
in Algorithm4 are simple. Anotheradvantageof Algorithm4 e) Set Γ(r+1) = (Γ(r))[1] and Θ˜(r+1)(x) = x[1] ⊗
is that it does not require the roots to be linearly independent. ΘP˜(r)(x).
In Algorithm 4, F(i)(wj) = 0 for j = 0,1,...,i − 1 and 5.3 Set Λ(x)= t ∆˜(2t)x[i].
γ(i+1) = F(i)(w ). If w ,w ,...,w are linearly dependent, i=0 i+t
i i 0 1 j
γ(j) =0 and hence w is ignored. So Algorithm 4 integrates Following AlgoPrithm 5, we propose a systolic RiBMA ar-
j j
detection of linearly dependency at no extra computational chitecture shown in Fig. 3, which consists of 3t+1 identical
cost. processingelements(BEs),whosecircuitryisshowninFig.4.
Essentially, Algorithm 4 breaks down evaluations of high The central control unit BCtrl processes b inside, generates
q-degree polynomials into evaluations of polynomials with q- the globalcontrolsignals ct(r) and Γ(r), and passes alongthe
degree as one. It gets rid of complex operations while main- coefficient Λ(r). The control signal ct(r) is set to 1 only if
0
taining the same total complexity of the algorithm. ∆˜(r) 6= 0 and k > 0. In each processing element, there are
0
two criticalpaths, bothofwhich consistof onemultiplierand
oneadderoverF .Onestartsfrom∆˜(r) andtheotherΘ˜(r) ,
IV. ARCHITECTURE DESIGN 2m i+1 i+1
while both end in ∆˜(r).
Aiming to reduce the storage requirementand total area as i
well as to improvethe regularity of our decoder architectures,
we furtherreformulatethe stepsin thedecodingalgorithmsof ∆˜i(r)
both Gabidulinand KK codes. Again, we assume the decoder
architectures are suitable for RLNC over F , where q is a ∆˜(r) D + × ∆˜(r)
q i i+1
power of two.
Γ(r) Γ(r)
ct(r) 1 0 ct(r)
A. High-Speed BMA Architecture
Toincreasethethroughput,regularBMAarchitectureswith ∆˜(r) xq × ∆˜(r)
0 0
shorter CPD are necessary. Following the approaches in [24],
D
wedeveloptwoarchitecturesbasedonAlgorithm3,whichare
analogous to the riBM and RiBM algorithms in [24]. Θ˜(r) Θ˜(r)
i i+1
In Algorithm 3, the critical path is in step 3.2(a). Note
that ∆r is the rth coefficient of the discrepancy polynomial Fig. 4. The processing element BEi (xq is a cyclic shift, and requires no
∆(r)(x) = Λ(r)(x)⊗S(x). By using Θ(r)(x) = B(r)(x)⊗ hardwarebutwiring)
S(x), ∆(r+1)(x) can be computed as
∆(r+1)(x)=Λ(r+1)(x)⊗S(x)
B. Generalized BMA
= (Γ(r))[1]Λ(r)(x)−∆ x[1]⊗B(r)(x) ⊗S(x)
r ThekeyequationofKKdecodingisessentiallythesameas
=h(Γ(r))[1]∆(r)(x)−∆ x[1]⊗Θ(r)(x) i (10) (3) but ω(x) has q-degree less than τ, instead of ⌊(d−1)/2⌋.
r
Actually, in KK decoding, we do not know the exact value
whichhasthestructureasstep3.2(c).Hencethisreformulation of τ before solving the key equation. But all we need is to
is more conducive to a regular implementation. determinethe maximumnumberofcorrectableerrorst′ given
Given the similarities between step 3.2(a) and (10), Λ(x) µ erasures and δ deviations, which is given by t′ = ⌊(d−
and∆(x)canbecombinedtogetherintoonepolynomial∆˜(x). 1 − µ − δ)/2⌋. Hence we need to generalize our previous
9
Λ Λ
0 t
0
BE0 ··· BEt−1 BEt ··· BE2t BE2t+1 ··· BE3t BCtrl
0
Fig.3. TheRiBMAarchitecture
BMA architectures in Section III-C for KK decoding, as in Algorithm7,hasthreekeydifferencesfromthepivotingarchi-
Algorithm 6. To apply Algorithm 6 to Gabidulin decoding, tecturein[41].First,Algorithm7isapplicabletomatricesover
we can simply use θ =µ+δ =0. any field. Second and more importantly, Algorithm 7 detects
singularityandcan beused forsingularmatrices. Thisfeature
Algorithm 6 (Generalized RiBMA).
is necessary since singular matricesoccurin the reductionfor
Input: S and θ
the RRE form and finding the root space. Third, Algorithm 7
Output: Λ(x)
is also flexible about matrix sizes, which are determined by
6.1 Initialize as follows: t′ = ⌊(d−1−θ)/2⌋, ∆˜(0)(x) = the variable numbers of errors, erasures, and deviations.
Θ˜(0)(x) = θ+2t′−1S x[i], ∆˜(0) = Θ˜(0) = 1,
i=θ i 2t′+t 2t′+t Algorithm 7 (Gaussian Elimination for Root Space).
6.2 FΓo(0r)r==1,0,a1n,dP..b.=,20t′.−1, Input: m×m matrices M ∈Fmq ×m, whose rows are eval-
uations of σ(x) over the normal basis, and B =I
a) Modify the combined polynomial: ∆˜(r+1)(x) = Output: Linearly independentroots of σ(x)
Γ(r)∆˜(r)(x)−∆˜(r)Θ˜(r)(x);
0 7.1 Set i=0.
b) Set b=b+1;
c) If ∆˜(r) 6=0 andb>0,set b=−b, Γ(r+1) =∆˜(r), 7.2 For j =0,1,...,m−1
0 0
and Θ˜(r)(x)=∆˜(r)(x); a) While M0,0 = 0 and j < m−i, shiftup(M,i)
d) Set ∆˜(r+1)(x)= 2t′+t−1∆˜(r+1)x[i], Θ˜(r)(x) = and shiftup(B,i).
2t′+t−1Θ˜(r) x[i]; i=0 i+1 b) IfM0,0isnotzero,eliminate(M),reduce(B,M),
i=0 i+1 P and i=i+1; Otherwise, shiftleft(M).
e) Set Γ(r+1) = (Γ(r))[1] and Θ˜(r+1)(x) = x[1] ⊗
PΘ˜(r)(x). 7.3 The last m−i rows of M are all zeros and the last
m−i rows of B are roots.
6.3 Set Λ(x)= t′ ∆˜(2t′)x[i].
i=0 i+t
The eliminate and shiftup operations are quite similar to
Compared withPAlgorithm 5, we replace t by t′. The vari- those in [41, Algorithm 2]. In eliminate(M), for all 0 ≤
able t′ makes it difficult to design regular architectures. By i < m and 0 ≤ j < m, M = M M −
i,j 0,0 i+1,(j+1)modm
carefully initializing ∆˜(0)(x) and Θ˜(0)(x), we ensure that the M M ifi<m−1,orM =M .
desired outputΛ(x) is always at a fixed position of ∆˜(2t′)(x), Thie+1p,r0oce0d,(uj+re1)rmedoudcme(B,M) essentialliy,j repea0ts,(je+l1im)minoadtme
regardless of µ + δ. Hence, the only irregular part is the withoutcolumnoperations.Forall0≤i<mand0≤j <m,
initialization. The initialization of Algorithm 6 can be done B = M B −M B if i < m− 1, otherwise
i,j 0,0 i+1,j i+1,0 0,j
by shifting in at most θ cycles. Hence we reuse the RiBMA B = B . In the shiftup(M,ρ) operation, the first row
i,j 0,j
architecture in Fig. 3 in the KK decoder and keep the same is moved to the (m−1−ρ)th row while the second to the
worse-case latency of 2t cycles. (m−1−ρ)th rows are moved up. That is, for 0 ≤ j < m,
M = M if i = m − 1 − ρ, and M = M for
i,j 0,j i,j i+1,j
C. Gaussian Elimination 0 ≤ i ≤ m−2−ρ. In the shiftleft operation, all columns
are cyclic shifted to the left except that the first column is
We need Gaussian elimination to obtain n-RRE forms as
moved to the last. In other words, for all 0 ≤ i < m and
well as to find root spaces. Furthermore, Gabidulin’s algo-
0 ≤ j < m, M = M . By adding a shiftleft
rithm in Algorithm 1 is essentially a smart way of Gaussian i,j i,(j+1)modm
operation,Algorithm7handlesbothsingularandnon-singular
elimination, which takes advantage of the properties of the
matrices while [41, Algorithm 2] only works on non-singular
matrix. The reduction (to obtain n-RRE forms) and finding
matrices. Since B is always full rank, the roots obtained are
the root space are Gaussian eliminations on matrices over F ,
q guaranteed to be linearly independent.
whileGabidulin’salgorithmoperatesonmatricesoverF .In
qm We can get the root space using Algorithm 7, and we can
this section, we focus on Gaussian eliminations over F and
q alsouse itinKK decodingto reducethereceivedvectorto an
Gabidulin’s algorithm will be discussed in Section IV-D. n-RRE form. But Algorithm 7 only provides Eˆ′, we need to
Forhigh-throughputimplementations,weadoptthepivoting extend it to obtain Lˆ′, as in Algorithm 8.
architecture in [41], which was developed for non-singular
matrices over F . It always keeps the pivot element on the Algorithm 8 (Gaussian Elimination for n-RRE Forms).
2
ˆ
top-left location of the matrix, by cyclically shifting the rows Input: N ×n matrix A and N ×m matrix y
and columns. Our Gaussian elimination algorithm, shown in Output: Lˆ′, Eˆ′, r′, and µ′
10
8.1 Set i=0, U′ and Lˆ as empty. In Gabidulin’salgorithm,there are two τ×τ matricesover
8.2 For each column j =0,1,...,n−1 F , A andQ. In step 1.2,it requiresonlyQ ’s to compute
qm i,0
a) While Aˆ0,0 = 0 and j < n − i, shiftup(Aˆ,i), thecoefficients.TocomputeQi,0in(6),itrequiresonlyQi−1,0
shiftup(y,i), shiftup(Lˆ′,i). and Qi−1,1. And for Qi,j in (6), it requires only Qi−1,j and
b) If Aˆ0,0 is not zero, eliminate(Aˆ), reduce(y,Aˆ), Qi−1,j+1. Recursively only those Qi,j’s where i+j <τ are
shiftup(Lˆ′,0), i=i+1. necessary.Actually,givenanyi,entriesQi,0,Qi+1,0,...,Qτ−1,0
c) Otherwise, shiftleft(Aˆ), append the first column canbecomputedwiththeentriesQi−1,0,Qi−1,1,...,Qi−1,τ−i.
of Aˆ to Lˆ′, set the top-rightelement of Lˆ′ to one, WithQ0,0,Q1,0,...,Qi−2,0,weneedtostoreonlyτ valuesto
and add j to U′. keeptrackofQ.HencewereducethestorageofQfromτ×τ
8.3 Set µ′ =n−i. The deviationsEˆ′ are given by the first m-bitregistersdownto τ. We cannotreducethe storageof A
µ′ rows of y. to τ(τ+1)/2becausewe havetouse the pivotingscheme for
8.4 Foreachcolumnj ∈U′,shiftup(Lˆ′,j)andshiftup(y,j). short critical paths.
8.5 The received vector r′ is given by y.
′
InAlgorithm8,weincorporatetheextractionofLˆ′,Eˆ′,and Ai,j Ai−1,j
′
r′ intoGaussianelimination.Soourarchitecturehasthesame Ai−1,j−1
worst-caselatencyasAlgorithm7andrequiresnoextracycles
toextractLˆ outofthen-RREform.Hencethethroughputalso x1q
ctau ctau
remains the same. i 1 0 i
Algorithm7isimplementedbytheregulararchitectureshown
× +
inFig.5,whichisatwo-dimensionalarrayofm×2mprocess- (q−1)/q (q−1)/q
A A
ing elements (GE’s); The leftmost m columns of processing i−1,i−1 i−1,i−1
elements correspond to M, and the rightmost m columns B. 0 D Ai,j
1
Algorithm 8 can be implemented with the same architecture
ctal ctal
withN×(n+m)GE’s;Theleftmostncolumnsofprocessing i o
ˆ
elements correspond to A, and the rightmost m columns y. A
The elements for Lˆ′ are omitted in the figure. The circuitry i+1,j+1
A
of the processing element GE is shown in Fig. 6. For row i, i,j
thecontrolsignalct forrowichoosesfromfiveinputsbased
i
on the operation: keeping the value, shiftleft, eliminate (or Fig.8. Theprocessing elementAEi,j
reduce), and shiftup (using the first row or the next row).
′
X q q
Aei,j A0,j Ai,j j i i+1
A0,0 A0,0 A(q−1)/q × 1
A A i−1,i−1 xq
i,0 1 0 × i,0
1
+ 2 D
0
+
Ai,j × D MUX Ai,j−1 Ai,j ctqi
ct ct × +
i i
A Ae
A0,j i+1,j i+1,j+1 Xj qi+1 qi′ qi
Fig.6. TheprocessingelementGEi,j Fig.9. Theprocessing elementQE
i
In our decoder, Gabidulin’s Algorithm is implemented by
D. Gabidulin’s Algorithm the regular architecture shown in Fig. 7, which include a tri-
In Gabidulin’s algorithm, the matrix is first reduced to a angular array of τ × τ AE’s a one-dimensional array of τ
triangularform.Ittakesadvantageofthepropertyofthematrix QE’s. Thecircuitryof the processingelementAE andQE
i,j i
such that it requires no division in the first stage. In the first is shown in Fig. 8 and9. The upperMUX in AE controlsthe
stage, we need to perform elimination on only one row. We output sending upward along the diagonal. Its control signal
useasimilarpivotingschemelikeAlgorithm7.Whenarowis ctau is 1 for the second row and 0 for other rows since we
i
reducedto have onlyonenon-zeroelement,a divisionis used update A one row in a cycle and we keep the pivot on the
to obtain one coefficient of X. Then it performs a backward upperleftcornerinStep1.1.ThecontrolofthelowerMUXin
elimination after getting each coefficient. Hence we introduce AEis0forworkingonStep1.1,and1forworkingonStep1.2.
abackwardpivotingscheme,wherethepivotelementisalways Similary the control of the MUX in QE is 0 for working on
at the bottom-rightcorner. Step 1.1, and1 forworkingonStep 1.2.Butin Step 1.1,only