Table Of ContentList Decoding of Matrix-Product Codes
from nested codes: an application to
Quasi-Cyclic codes∗
2
1
0 Fernando Hernando1
2
DepartmentofMathematics, UniversidadJaumeI,
n
CampusRiuSec,12071, Castello´ndelaplana,Spain
a [email protected]
J
Tom Høholdt
0
3
DTU-Mathematics,Technical UniversityofDenmark,
Matematiktorvet, Building303,2800Kgs. Lyngby,Denmark
] [email protected]
T
I Diego Ruano
.
s
c DepartmentofMathematical Sciences,AalborgUniversity,
[ Fr. Bajersvej7G,9920-AalborgØst,Denmark
[email protected]
1
v
Abstract
7
9 A list decoding algorithm for matrix-product codes is provided when
3 C ,...,C are nested linear codes and A is a non-singular by columns
1 s
6
matrix. Weestimatetheprobabilityofgettingmorethanonecodewordas
.
1 output when the constituent codes are Reed-Solomon codes. We extend
0 this list decoding algorithm for matrix-product codes with polynomial
2 units, which are quasi-cyclic codes. Furthermore, it allows us to consider
1 uniquedecoding for matrix-product codes with polynomial units.
:
v
i
X 1 Introduction
r
a
Matrix-productcodes,[C ···C ]·A,areageneralizationofseveralclassiccodes
1 s
constructions of codes from old ones [2, 15]. For instance, they extend the
(u|u+v)-construction. An algorithm for unique decoding when the codes are
nested, C ⊃ ··· ⊃ C , and A has a certain property, called non-singular by
1 s
columns,wasprovidedin[8]. Thealgorithmdecodesuptohalfoftheminimum
∗This work was supported in part by the Danish National Research Foundation and the
NationalScienceFoundationofChina(GrantNo.11061130539)fortheDanish-ChineseCenter
forApplications ofAlgebraicGeometry inCodingTheoryand Cryptography, bythe Claude
ShannonInstitute,ScienceFoundationIrelandGrant06/MI/006andbySpanishMECGrant
MTM2007-64704. 2000 Mathematics Subject Classification. Primary: 94B05; Secondary:
94B35. Keywords: LinearCode,Matrix-ProductCode,ListDecoding, Quasi-CyclicCode.
1INSPIREfellowfundingreceivedfromtheIrishResearchCouncilforScience,Engineering
andTechnology.
1
distance, assuming that we have a decoding algorithm for C that decodes up
i
to half of its minimum distance, for every i.
List decoding was introduced by Elias [5] and Wozencraft [16]. The list
decoderisarelaxationoveruniquedecodingthatallowsthedecodertoproduce
a list of codewords as answers. It can uniquely decode beyond half of the
minimum distance in some cases or to produces a list of codewords.
In 1997, Sudan presented a polynomial time algorithm for decoding low
rate Reed-Solomon codes beyond the classical d bound. Later [7], Guruswami
2
and Sudan provided a significantly improved version of list decoder which can
correctcodes ofany rates. Recently, Lee andO’Sullivanprovidea list decoding
algorithm based on the computation of a Gr¨obner basis of a module [13] and
BeelenandBranderprovideanalgorithmthathaslinearcomplexityinthecode
length [1].
Inthispaperweconsideralistdecodingalgorithmformatrix-productcodes
which is an extension of the algorithm in [8]. The algorithm in [8] assumes a
knowndecodingalgorithmforeveryconstituentcodeC thatdecodesuptohalf
i
of its minimum distance, for this algorithm, we assume that the decoding algo-
rithm is a list-decoding algorithm. Moreover,it is also requiredthat C ,...,C
1 s
are nested and A is non-singularby columns. The extensionis natural, but, we
believe, it is non-trivial task since we had to modify the algorithmto deal with
lists of codewords,compute the error bound τ and prove the correctness of the
algorithm, among others.
Matrix-Productcodes are generalizedconcatenated codes [2, 4], which have
an efficient decoding algorithm [6]. However, this algorithm cannot be success-
fully applied if the matrix A is small, as it is in practice for Matrix-product
codes (see Remark 3.4).
Theprobabilityofgettingmorethanonecodewordasoutputofalistdecod-
ingalgorithmforReed-Solomoncodeswasboundedin[14]. Insection4,weuse
thiscomputationtoestimateanupperboundoftheprobabilityofgettingmore
than one codeword as output, when C ,...,C are Reed-Solomon codes. The
1 s
algorithminsection3canbecomecomputationallyintense,anoptimalsituation
arises considering s=l =2.
In section 5 we extend the algorithm in section 3 for matrix-product codes
with polynomial units [9], which are quasi-cyclic codes. Quasi-cyclic codes be-
cameimportantafteritwasshownthatsomecodesinthisclassmeetamodified
Gilbert-Varshamovbound[10], howeverthereareno generalfastalgorithmsfor
decoding them. In [9], many of these codes with good parameters were ob-
tained. Using list decoding of matrix-product codes with polynomial units we
can uniquely decode these codes up to the half of the minimum distance.
2 Matrix-Product Codes
A matrix-product code is a construction of a code from old ones.
Definition 2.1. LetC ,...,C ⊂Fm be linearcodesoflengthm andamatrix
1 s q
A = (a ) ∈ M(F ,s × l), with s ≤ l. The matrix-product code C =
i,j q
[C ···C ]·A is the set of all matrix-products [c ···c ]·A where c ∈C is an
1 s 1 s i i
m×1 column vector c =(c ,...,c )T for i=1,...,s. Therefore, a typical
i 1,i m,i
codeword c is
2
c a +···+c a ··· c a +···+c a
1,1 1,1 1,s s,1 1,1 1,l 1,s s,l
c= ... ... ... . (1)
c a +···+c a ··· c a +···+c a
m,1 1,1 m,s s,1 m,1 1,l m,s s,l
s
Clearlythei-thcolumnofanycodewordisanelementoftheform a c ∈
j=1 j,i j
Fm, therefore reading the entries of the m×l-matrix above in column-major
q
P
order, the codewords can be viewed as vectors of length ml,
s s
c= a c ,..., a c ∈Fml. (2)
j,1 j j,l j q
j=1 j=1
X X
FromtheaboveconstructionitfollowsthatageneratormatrixofC isofthe
form:
a G a G ··· a G ··· a G
1,1 1 1,2 1 1,s 1 1,l 1
a G a G ··· a G ··· a G
2,1 2 2,2 2 2,s 2 2,l 2
G= . . . . ,
. . . .
. . ··· . ··· .
a G a G ··· a G ··· a G
s,1 s s,2 s s,s s s,l s
whereG isageneratormatrixofC ,i=1,...,s. Moreover,ifC isa[m,k ,d ]
i i i i i
codethenonehasthat[C ···C ]·AisalinearcodeoverF withlengthlmand
1 s q
dimension k =k +···+k if the matrix A has full rank and k<k +···+k
1 s 1 s
otherwise.
Letus denote by R =(a ,...,a ) the element ofFl consisting ofthe i-th
i i,1 i,l q
row of A, for i=1,...,s. We denote by D the minimum distance of the code
i
C generated by hR ,...,R i in Fl. In [15] the following lower bound for the
Ri 1 i q
minimum distance of the matrix-product code C is obtained,
d(C)≥min{d D ,d D ,...,d D }, (3)
1 1 2 2 s s
where d is the minimum distance of C . If C ,...,C are nested codes, C ⊃
i i 1 s 1
···⊃C , the previous bound is sharp [8].
s
In [2], the following condition for the matrix A is introduced.
Definition 2.2. [2] Let A be a s×l matrix and A be the matrix consisting of
t
the first t rows of A. For 1 ≤ j < ··· < j ≤ l, we denote by A(j ,...,j ) the
1 t 1 t
t×t matrix consisting of the columns j ,...,j of A .
1 t t
A matrix A is non-singular by columns if A(j ,...,j ) is non-singular for
1 t
each 1 ≤ t ≤ s and 1 ≤ j < ··· < j ≤ l. In particular, a non-singular by
1 t
columns matrix A has full rank.
Moreover, if A is non-singular by columns and C ⊃ ··· ⊃ C , we have
1 s
d(C)=min{ld ,(l−1)d ,...,(l−s+1)d } [8].
1 2 s
In[8]werepresentedadecodingalgorithmforthe matrix-productcodeC =
[C ···C ] · A ⊂ Fml, with A non-singular by columns and C ⊃ ··· ⊃ C ,
1 s q 1 s
assuming that we have a decoding algorithm for C , for i = 1,...,s. The
i
algorithmin[8]decodesuptohalfofthe minimumdistance. Innextsectionwe
provide a list decoding algorithm for such codes, assuming that we have a list
decoding algorithm for C , i=1,...,s.
i
3
3 List Decoding Algorithm for matrix-product
codes
Let C ⊂Fn and τ >1. For r ∈Fn, a list decoding algorithm with error bound
q q
τ provides a list with all codewords in C that differ from r in at most τ places.
If τ ≤⌊d−1⌋, it will result into unique decoding.
2
We present a list decoding algorithm for a class of matrix-product codes, it
is an extension of [8, Algorithm 1]. Namely, we consider s nested linear codes
C ,...,C ⊂ Fm and a non-singular by columns matrix A ∈ M(F ,s × l),
1 s q q
where s≤l. We provide a list decoding algorithm for the matrix-product code
C = [C ···C ]· A ⊂ Fml, assuming that we have a list decoding algorithm
1 s q
LDC for C with error bound τ . In particular, each LDC answers an empty
i i i i
list if there is no codeword in C within distance τ of the received word.
i i
Our list algorithm for C decodes up to
τ =min{lτ +(l−1),(l−1)τ +(l−2),...,(l−s+1)τ +l−s}. (4)
1 2 s
We first describe the main steps in our decoding algorithm. The algorithm
is outlined as a whole in procedural form in Algorithm 1.
Consider the codeword c = ( s a c ,..., s a c ), where c ∈ C ,
j=1 j,1 j j=1 j,l j j j
for all j. Suppose that c is sent and that we receive p = c+e, where e =
(e ,e ,...,e )∈Fml isanerrorvePctor. WedenotePbyp = s a c +e ∈Fm
1 2 l q i j=1 j,i j i q
the i-th block of p, for i = 1,...,l. Let {i ,...,i } ⊂ {1,...,l} be an ordered
1 s
P
subset of indices. We now also suppose that e satisfies the extra property that
wt(e )≤τ for all j ∈{1,...,s}. (5)
ij j
Since C ⊃ ··· ⊃ C , each block s a c of c is a codeword of C .
1 s j=1 j,i j 1
Therefore, we decode the i -th block p of p using LDC and we obtain a list
L . Since wt(e )≤τ , we1have s ai1Pc ∈L . In pra1ctice we do not know
1 i1 1 j=1 j,i1 j 1
s
whichoneoftheelementsinL is a c ,thereforeweshouldconsiderthe
1 P j=1 j,1 j
following computations for every element in L . Assume now that we consider
1
s P s
a c ∈L ,henceweobtaine =p − a c andwecaneliminate
j=1 j,i1 j 1 i1 i1 j=1 j,i1 j
c in every other block (although we do not know c ) in the following way: we
1 1
cPonsider a new vector p2) ∈Fml with componPents
q
s
a
p2) =p − 1,i (p −e )= a2)c +e , for i6=i ,
i i a i1 i1 j,i j i 1
1,i1 j=2
X
where a2) = a − a1,i a , and p2) = p −e . Since A is a non-singular
j,i j,i a1,i1 j,i1 i1 i1 i1
by columns matrix, the elements of the first row of A are non-zero, and so the
denominator a is non-zero.
1,i1
Since C ⊃ ··· ⊃ C , we notice that the i-th block of p2) is a codeword
2 s
of C plus the error block e , for i ∈ {1,...,s}\ {i }. We now decode the
2 i 1
i -th block p2) = s a2) c +e of p2) using LDC and we obtain a list L .
2 i2 j=2 j,i2 j i2 2 2
Sincew(e )≤τ ,wehave s a2) c ∈L . Inpracticewedonotknowagain
i2 2P j=2 j,i2 j 2
whichoneoftheelementsinL is s a2) c ,thereforeweshouldconsiderthe
P2 j=2 j,i2 j
following computations for every element in L . Assume now that we consider
2
s a2) c ∈ L , hence we obtaPin e and, as before, we can eliminate c in
j=2 j,i2 j 2 i2 2
P
4
every other block (although we do not know c ) as follows: we consider a new
2
vector p3) ∈Fml with components
q
a2) s
p3) =p2)− 2,i (p2)−e )= a3)c +e , for i6=i ,i ,
i i a2) i2 i2 j,i j i 1 2
2,i2 Xj=3
where a3) =a2) − a22),i a2) , p3) =p2) and p3) =p2)−e .
j,i j,i a2) j,i2 i1 i1 i2 i2 i2
2,i2
Notice that the i-th blockof p3) is a codewordofC plus the errorblock e ,
3 i
for i∈{1,...,s}\{i ,i }.
1 2
Then we iterate this process, defining pk) for k = 3,...,s, and decoding
the i -th block using LDC . In this way, we obtain the error blocks e , and
k k i
s
the corresponding codeword blocks a c , for i∈{i ,...,i }. The vector
j=1 j,i j 1 s
( s a c ,..., s a c ) formed from these s decoded blocks is equal to
j=1 j,i1 j j=1 j,is j P
theproduct[c ···c ]·A(i ,...,i ),whereA(i ,...,i )isthes×s-submatrixof
1 s 1 s 1 s
P P
Aconsistingofthecolumnsi ,...,i . Sincethismatrixisfullrank,wecannow
1 s
easilycompute c ,...,c by invertingA(i ,...,i ) orsolving the corresponding
1 s 1 s
linearsystem. Finallywerecovertheremainingl−scodewordblocks“forfree”
(i.e. no decoding procedureis involvedforthese blocks)by simply recomputing
the entire codeword c=[c ···c ]·A=( s a c ,..., s a c ), since we
1 s j=1 j,1 j j=1 j,l j
know the c ’s and the matrix A.
j
P P
k)
Foreacheliminationstepintheaboveprocedure,itisnecessarythata 6=
k,ik
0, for each k =2,...,s, to avoid zero division. We claim that this follows from
the non-singular by columns property of A, exactly in the same way as in [8].
Let A1) = A. The matrix Ak) = (ak))∈ M(F ,s×l),k = 2,...,s, is obtained
i,j q
recursivelyfromAk−1) byperformingthefollowingl−(k−1)elementarycolumn
operations:
ak−1)
column (Ak))=column (Ak−1))− k−1,i column (Ak−1)),
i i ak−1) ik−1
k−1,ik−1
foreachi∈/ {i ,...,i }. Theseoperationsintroducel−(k−1)additionalzero
1 k−1
elementsinthek−1-throwofAk) ateachiteration. HencethesubmatrixofAk)
givenbythefirstkrowsandthei ,...,i columns,isatriangularmatrix(inthis
1 k
case, a column permutation of a lower triangular matrix) whose determinant is
ak) ···ak) . SinceAisnon-singularbycolumns,thissubmatrixisnon-singular.
1,i1 k,ik
It follows that the determinant is non-zero, and therefore ak) 6=0.
k,ik
The procedure described above will generate a list that includes the sent
word, if for a given choice of indices {i ,...,i } ⊂ {1,...,l}, each error block
1 s
satisfies wt(e ) ≤ τ , for all j = 1,...,s. The output’s procedure may not
ij j
include the sent word if wt(e )>τ for some j.
ij j
In the previous description, we have only shown the computations for one
ofthe differentchoicesthatthe listdecoderalgorithmsC ,fori=1,...,s,may
i
give us. However, if #L > 1 then we should consider a different word p2) for
1
every ℓ ∈ L . One can see how this tree is created for every element in L in
1
line 8 of Algorithm1. If a decoder LDC outputs anempty list, for all possible
j
choices in L , then we consider another ordered subset of indices, and start
j−1
the procedure again.
5
We now prove that for every error vector e with wt(e) ≤ τ there exists a
good set of indices {i ,...,i } ⊂ {1,...,l} satisfying condition (5). We should
1 s
repeat the procedure described above, with every ordered subset of indices and
collect all the decoded words,in order to be sure that the “good”set of indices
is considered.
Theorem3.1. LetC bethematrix-productcode[C ···C ]·A,whereC ⊃···⊃
1 s 1
C and A is a non-singular by columns matrix. Let e=(e ,e ,...,e )∈Fml be
s 1 2 l q
an error vector with wt(e) ≤ τ (see (4)). Then there exists an ordered subset
{i ,...,i }⊂{1,...,l} satisfying wt(e )≤τ , for all j ∈{1,...,s}.
1 s ij j
Proof. We claimthatthere existsi suchthatwt(e )≤τ . Suppose thatthere
1 i1 1
isnoi ∈{1,...,l}withwt(e )≤τ ,thatis,wt(e )≥τ +1,foralli=1,...,l.
1 i1 1 i 1
This implies that
wt(e)=wt(e )+···+wt(e )≥lτ +l >τ
1 l 1
which contradicts our assumption.
Letusassumethatthepropertyholdsforasubset{i ,...,i }⊂{1,...,l}
1 j−1
of size j − 1 < s. We now prove it holds for a subset of size j. Suppose
that there is no i with wt(e ) ≤ τ , that is, wt(e ) > τ + 1, for all i ∈
j ij j i j
{1,...,l}\{i ,...,i }. This implies that
1 j−1
j−1 l
wt(e) ≥ wt(e )+ wt(e )
ik ik
k=1 k=j
X X
j−1
> wt(e )+(l−j+1)τ +(l−j+1)
ik j
k=1
X
≥ (l−j+1)τ +(l−j+1)>τ
j
which contradicts our assumption and the result holds.
Summarizing,wecannowformulateourdecodingalgorithmforC =[C ···C ]·
1 s
A ⊂ Fml, where C ⊃ ··· ⊃ C and A is a non-singular by columns matrix, in
q 1 s
procedural form in Algorithm 1.
Corollary 3.2. If wt(e) ≤τ then c is in the list given as output of Algorithm
1. Hence, the algorithm described in this section is a list decoding algorithm
with error bound τ, i.e, L={c∈C |wt(p−c)≤τ}.
Proof. By 3.1, there exists an ordered subset {i ,...,i } ⊂ {1,...,l} satis-
1 s
fying wt(e ) ≤ τ , for all j ∈ {1,...,s}. Therefore, c ∈ L and c =
s ij js ij j
( a c ,..., a c ) ∈ L. Furthermore, all the words at distance τ
j=1 j,1 j j=1 j,l j
from the received word are included in the output list as well.
P P
Example 3.3. Consider the matrix-product codes with matrix A of the form
1 1
A= ,
0 1
(cid:18) (cid:19)
and C ⊃ C Reed-Solomon Codes over F with parameters [15,10,6] and
1 2 16
[15,4,12], respectively. Therefore, the code C = [C C ] · A has parameters
1 2
6
Algorithm 1 List decoding algorithm for C =[C ···C ]·A
1 s
Input: Received word p = c+e with c ∈ C and wt(e) ≤ τ. C ⊃ ··· ⊃ C
1 s
nested codes and A a non-singular by columns matrix. Decoder LDC for
i
code C , i=1,...,s.
i
Output: List of all codewords that differ from p in at most τ places.
1: p′ =p; A′ =A; Dec={};
2: for {i1,...,is}⊂{1,...,l} do
3: p=p′; A=A′; U′ ={p};
4: for j =1,...,s do
5: U =U′; U′ ={};
6: for u in U do
7: L=LDCj(uij);
8: for ℓ in L do
9: tmp=(0,...,0)∈Fm;
10: for k =j+1,...,s do
11: tmpik =uik − aajj,,iikjℓ;
12: end for
13: U′ =U′∪{tmp};
14: end for
15: end for
16: if U′ ={} then
17: Break the loop and consider another i1,...,is in line 2;
18: end if
19: for k =j+1,...,s do
20: columnik(A)=columnik(A)− aajj,,iikjcolumnij(A);
21: end for
22: end for
23: for u in U′ do
24: Obtain (c1,...,cs) from ui1,...,uis;
25: p=[c1···cs]·A; (see (1) and (2))
26: if wt(p−p′)≤τ then
27: Dec=Dec∪{p};
28: end if
29: end for
30: end for
[30,14,12]. We have error bounds τ = 3,τ = 7, for C and C respectively,
1 2 1 2
using the list decoding algorithm in [1] or in [13] with multiplicity v = 4 (see
next section for further details). Therefore, the error bound for Algorithm 1 is
τ =7. Note that the error correction capability of C with [8] is only t=5.
Let c = (0,0) be the sent word and p = (α2x+αx5 +α5x6 +α14x13,α5x2 +
α7x6+α8x10) the received word, i.e. wt(e)=7.
• We consider the ordered set of indices {1,2}. So, we decode p = α2x+
1
αx5+α5x6+α14x13 with the list decoding algorithm for C : we obtain
1
p2) =α2x+αx5+α5x6+α14x7+α10x13+α5x14.
1
Then we compute p2) = p −p2) = α2x+α5x2 +αx5 +α5x6 +αx7 +
2 2 1
7
α8x10+α10x13+α5x14 anddecodeitwith the listdecodingalgorithmfor
C . However, we get an empty list as output and we do not consider any
2
codeword for the final list.
• We consider now the ordered set of indices {2,1}. Therefore, we decode
p = α5x2 +α7x6 +α8x10 with the list decoding algorithm for C . We
2 1
obtainas output p2) =0. Thus, we compute p −p2) =p and we decode
2 1 2 1
it with the list decoding algorithm for C . We have obtain 0. Therefore,
2
we deduce that the sent codeword is (0,0).
Remark 3.4. Matrix-product codes are generalized concatenated codes [2].
There is an efficient decoding algorithm for generalized concatenated codes [4],
the cascaded decoding of multilevel concatenations. In [6], Guruswami and
Rudra generalize this algorithm to a list decoding algorithm for generalized
concatenated codes.
The latter are defined as follows: consider s outer codes, say Cojut over Fqaj
withparameters(N,K ,D )forj =0,...,s−1. LetC =C0 ×···×Cs−1 =
j j out out out
{(c0,...,cs−1) | cj ∈ Cj ,j = 0,...,s−1}, understanding cj ∈ Cj as a row
out out
vector. Thus,a typicalelement c∈C is a s×N matrix, we denote by c the
out k
k-th column of c, for k =0,...,N −1.
One also considers a inner code C over F and a one-to-one map ψ from
in q
Fqa0 ×···×Fqas−1 to Cin that maps (i0,...,is−1) ∈ Fqa0 ×···×Fqas−1 to a
codeword ψ(i ,...,i ) in C .
0 s−1 in
A generalized concatenated code V of order s is the set
V ={ψ(c ),...,ψ(c )|(c ,...,c )∈B}.
0 N−1 0 N−1
GuruswamiandRudras’s algorithm[6] works as follows (we do not attempt
to write the algorithm, just to describe its main idea): let R ∈ M(a +···+
0
a × N,F ) be the received word. For j = 1,...,s −1 consider the code
s−1 q
Cj generated by all the rows of the generator matrix in C except the first
in in
a +···+a ,sayGj ,whereC0 =C . Thealgorithmassumestheexistence
0 j−1 in in in
of list decodable algorithms for Cj and list recovery algorithm for Cj . It is
in out
also assumedthat the list decoding algorithmfor Cj returns a list of messages
in
while the list recovery algorithm for Cj returns a list of codewords.
out
Inthe firstroundoneapplies,for i=0,...,N−1,a listdecoding algorithm
of C0 to ψ(c ) obtaining a list S0. For i = 0,...,N −1 and for each element
in i i
in S0, one can recover a message c˜ and project it into the first component,
i i
obtaining a list T0. Then it applies list recovery algorithm of C0 to {T0}
i out i i
obtaining a list L which in particular contains c0.
0
In the second round we proceed as follows, for each c ∈ L , c ∈ M(a ×
0 0
N,F ), consider the matrix cˆ∈M(a +···+a ×N,F ) which is the matrix
q 0 s−1 q
of zeroes with c in the first a rows. One may subtract R = R−(G0 )tcˆ, i.e.,
0 in
if c = c0 we are cancelling it from the received word. Thus we have a new
received wordR that should be decode with the generalized concatenated code
with outer codes C1 ,...,Cs−1 and inner code C1 . Since the number of outer
out out in
codes has dropped by one, repeating this process s-times one can successfully
list decode the original generalized concatenated code.
LetC be alinearcodeoverF withparameters[n,k,d]andgeneratorma-
in q
trixA,andletCj bealinearcodeoverF withparameters[N,K ,D ]forj =
out q j j
8
0,...,k−1. Consider ψ the encoding linear map of C , i.e., ψ(i ,...,i )=
in 0 k−1
(i ,...,i )A. ThenthegeneralizedconcatenatedcodeV isequaltothematrix
0 k−1
product code [C1 ,...,Cs ]·A. Since a matrix-product code [C ,...,C ]·A,
out out 1 s
with A non-singular by columns, is a generalized concatenated code one might
use the algorithm in [6] for matrix-product codes as well. However, the algo-
rithm in [6] can not be successfully applied for matrix-product codes because
the inner code has generator matrix A, that is a small matrix (in practice).
Theorem3.1guaranteesthe existence ofagood setofindices,i.e., satisfying
wt(e ) ≤ τ , for all j ∈ {1,...,s}. Hence, in the worst case, we may have
ij j
to consider s! ℓ iterations. However, in average we will consider much fewer
s
iterations. Given a fix set of ordered indices S = {i ,...,i }, we will estimate
1 s
(cid:0) (cid:1)
for how many errorpatterns of weightτ, one has a good set ofindices. In other
words, what is the probability that a set of indices {i ,...,i } is a good in the
1 s
worst case, i.e. when τ errors occur.
Proposition 3.5. If τ errors occur the probability that a fix set of indices
{i ,...,i } verifies that wt(e )≤τ for all j ∈{1,...,s} is:
1 s ij j
Pτa11=0Pma2i=n0{τ−a1,τ2}···Pamsi−n1{=τ−0Psj=−12aj,τs−1}(cid:0)am1(cid:1)(cid:0)am2(cid:1)···(cid:0)asm−1(cid:1)(cid:0)τm−(Pℓ−sj=−s+111a)j(cid:1)
(cid:0)mℓ(cid:1)
τ
Proof. Let us compute the different error vectors e with weight τ that allow us
to have a good set of indices. If wt(e ) ≤ τ , then we may have 0 ≤ a ≤ τ
i1 1 1 1
errors in block i , and there are m possibilities for having a errors in block
1 a1 1
i . Assuming that a errorsoccurredin the block i , we may have wt(e )≤τ
1 1 (cid:0) (cid:1) 1 i2 2
ifandonlyiftherearea errorsintheblocki ,where0≤a ≤min{τ−a ,τ },
2 2 2 1 2
sincewt(e)=τ. Hence,thereare m differentpossibilitiesforhavinga errors
a2 2
in the secondblock. Repeating this argumentfor the first s−1 blocks,we may
haveτ− s−1a errorsintheℓ−s(cid:0)+1(cid:1)remainingblocks(includingblocki )and
j=1 j s
therefore there are m(ℓ−s+1) possibilities for the remaining blocks. Overall
P τ−Psj=−11aj
we have
(cid:0) (cid:1)
τ1 min{τ−a1,τ2} min{τ−Psj=−12aj,τs−1} m m m m(ℓ−s+1)
··· ···
aX1=0 aX2=0 asX−1=0 (cid:18)a1(cid:19)(cid:18)a2(cid:19) (cid:18)as−1(cid:19)(cid:18)τ − js=−11aj(cid:19)
P
error patterns that make {i ,...,i } a good set of indices. The result holds
1 s
since there are mℓ error vectors of weight τ.
τ
In Example(cid:0)3.3(cid:1)we have s = l = 2 and τ = 3. Therefore the probability
1
that either {1,2} or {2,1} is a good set of indices is
15 15 + 15 15 + 15 15 + 15 15 1
0 7 1 6 2 5 3 4 = .
30 2
(cid:0) (cid:1)(cid:0) (cid:1) (cid:0) (cid:1)(cid:0) (cid:1) 7 (cid:0) (cid:1)(cid:0) (cid:1) (cid:0) (cid:1)(cid:0) (cid:1)
Finally we consider the complex(cid:0)ity(cid:1)of alogithm 1.
Theorem 3.6. Let LDC ,...,LDC be the list decoding algorithms consid-
1 s
ered in Algorithm 1 with error bounds τ ,...,τ , respectively, and that output
1 s
9
a list with size bounded by D ,...,D . We denote by R the complexity of the
1 s i
algorithm LDC . Then, algorithm 1 has complexity
i
s i−1
ℓ
O s! (D + ( D )R ) .
1 j i
s
(cid:18) (cid:19) i=2 j=1
X Y
Proof. In the worst case, one should consider every ordered set of s elements
within the l possible indices,thatis s! ℓ orderedsets. Fora fix orderedset, we
s
run LDC which yields a list of size at most D , in the worst case. For each
1 1
(cid:0) (cid:1)
element in this list we run LDC producing a list of size D , in the worst case.
2 2
HencewewillhaveD D wordsthatshouldbedecodedwithLDC . Repeating
1 2 3
thisprocess,intheworstcase,wewilldecodeD ···D wordswithLDC .
1 s−1 s
4 List decoding of Matrix-product codes from
Reed-Solomon codes with small s
Thepreviousalgorithmcanbecomecomputationallyintensiveasthenumberof
blocks l, the number of blocks that we may need to decode at each iteration s
and the error bounds τ ,...,τ , increase. Therefore, there are two interesting
1 s
situations: consideringfew blocks and considering codes and errorbounds such
that there is a small probability of getting a list with more that one element as
output of the list decoding algorithms LDC .
i
LetusconsiderthattheconstituentcodesC ⊃···⊃C areReed-Solomon.
1 s
Thisfamilyofcodesisespeciallyinterestinginthis settingfortworeasons,con-
sideran[m,k,d]Reed-Solomoncode,thereisanefficientlist-decodingalgorithm
[7] for decoding up to τv errors, with multiplicity v ∈N, which is computed as
follows
l
τv =m− v −1, where
v
(cid:22) (cid:23)
m v+1 (r −1)(k−1)
l = 2 + v and r is calculated so that
v v
r 2
$ (cid:0)v (cid:1) %
r m v+1 r +1
v ≤ 2 < v .
2 k−1 2
(cid:18) (cid:19) (cid:0) (cid:1) (cid:18) (cid:19)
In particular, one has the algorithm in [13] with complexity O(D4vm2) and
the one in [1] with complexity O(D4vmlog2mloglogm), where D is the list
size, v the multiplicity and m the length of the code. Notice that, by fixing v,
we fix τv and bound the list size D ≤ l /(k−1). Thus, one may obtain the
v
complexity of algorithm 1 as a function of the multiplicities and the length of
the constituent Reed-Solomon codes by Theorem 3.6.
Furthermore,onehasaboundforthetheprobabilityp (C)thattheoutput
τv
of Guruswami-Sudan’s algorithm for the code C with error bound τv has more
than 1 codeword[14], giventhat at mostτv errorshaveoccurredandassuming
that all the error patterns have the same probability. It turns out that this
probability may be very small in practice, for example a Reed-Solomon code
over F with parameters [64,20,45]and τ1 =23, this probability is 10−25.
26
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