On the non-existence for quantum LDPC codes of type IEEE802.16e with rates 1/2 and 2/3B

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by  Manabu Hagiwara

On the non-existence for quantum LDPC codes of
type IEEE802.16e with rates 1/2 and 2/3B
Manabu HAGIWARA Hideki IMAI
Research Center for Information Security (RCIS), Advanced Industrial Science and Technology (AIST)
National Institute of and Chuo University,
Advanced Industrial Science and Technology (AIST), 1-13-27 Kasuga, Bunkyo-ku,
Akihabara-Daibiru 11F, Tokyo, Japan.
8 1-18-13 Sotokanda, Chiyoda-ku, Tokyo, Japan. Email: [email protected]
0 Email: [email protected]
0
2

n
a I. INTRODUCTION and the Gaussian approximation method [10], [11].
J
Our design of irregular LDPC codes is based the design
1 In this paper, we discuss a construction of CSS codes written in the standardization of IEEE802.16e. It is need-
1
derived from pairs of practical irregular LDPC codes. Inter- less to say that irregular LDPC codes which are chosen

] section studies between quantum error-correcting codes and in IEEE802.16e are designed well-considerably and show
h LDPC codes are current hot research topics [1], [3], [4], [5], high error-correcting performance with practical length. Our
p
[6], [7], [8], [9]. One of aims of quantum error-correcting research has tried to make a CSS code with a pair of LDPC
-
t code theory is to construct quantum error-correcting codes codes of type IEEE802.16e. To our regret, we proved that it
n
a with small length and high error-correctingperformance.The was impossible to construct a CSS code if one of classical
u reason why LDPC codes are chosen is their high error- codes was of type IEEE802.16e with rate 1/2 and 2/3B. We
q correcting performances as classical error-correcting codes. would like to report the discussion on its impossibility in this
[ In fact, LDPC codes have almost achieved a theoretical paper. This is the first paper to analyze the possibility of a
2 error-correction limit, called a Shannon limit, with various CSS code construction by using two irregular LDPC codes
v conditions,forexample,coderate,codelength,communicatin which are practically useful.
1 channel, and others. This fact induces naturally derived the
6 quantum error-correcting codes. II. PRELIMINARIES
3
1 In the researches, we encounter a problems to hold a A. CSS codes
1. condition, which we call it the “twisted condition”, for a pair Denote the binary field by F2. CSS (Calderbank-Shor-
of classical codes to derive a quantum code. The condition
0 Steane) codes are quantum codes constructed by a pair of
8 requires a geometrically orderly structure to two classical classical linear codes C and D which satisfy the following
0 code spaces. More explicitly, the dual code of one of two
condition (T), called twisted relation:
: codes must be contained in the other code. On the other
v (T) D⊥ ⊂C,
i hand, a complicated structure, i.e. a random structure, is
X where D⊥ is the dual code of D. The dual code D⊥ of D is
suitable for achieving high error-correcting performance. The
r construction theory of classical codes for a CSS code is to defined by the following:
a
satisfy conflicting,orderlyandcomplicated,requirements.We D⊥ :={d′|d×d′T =0,∀d∈D},
re-write the two conditionsin terms of parity-check matrices.
Thetwistedconditionimpliestheorthogonalityoftheirparity- where d′T is the transposed vector of d′. A function h,i,
check matrices. A complex structure implies that non-zero defined by hx,yi := x×yT, gives an inner product over the
elements in the parity-check matrices should be irregularly binaryfieldF2.ThedualcodeD⊥isregardedasanorthogonal
arranged. space of D via h,i. Note that D⊥ is equal to a linear code
The previous researches on quantum LDPC codes have generatedby a parity-checkmatrix HD of D. In other words,
focused on quantum regular LDPC codes [1], [3], [7], [8] D⊥ ={xH |x∈Fm},
as same as that of classical LDPC codes. But now, a main D 2
stream of classical LDPC code researches is a construction where m is the column size of H .
D
method for irregular LDPC codes. The crucial reason why Details of the construction of a “quantum” CSS code by
the main research stream has changed is error-correcting such a pair of “classical” codes C and D is written in [13].
performance. In fact, the performances of irregular LDPC Remark 2.1: In general, a pair of linear codes forming a
codes are extremely better than those of regular LDPC codes. CSS code is denoted by C and C⊥. For simplicity, we use
1 2
This fact is theoretically proved by density-evolution theory C (resp. D) instead of C (resp. C⊥).
1 2
B. Quasi-Cyclic LDPC Codes ThemodelmatrixH associatedto H isa matrixofsize 3-by-
5:
A quasi-cyclic LDPC code with circulant matrices (QC-
∞ 0 1 0 ∞
LDPC code)isdefinedbya binarymatrixH ofsizem-by-n,  
H= 0 0 0 0 0 .
where n is the length of the code and m is the number of
 1 0 1 ∞ 0 
parity check bits in the code [2]. The number of systematic
Denote a set {0,1,...,z −1}∪{∞} by [z]∞. A model
bits is k =n−m. The matrix H is defined as:
matrixisamatrixover[z]∞.We defineaoperator−on[z]∞.
P0,0 P0,1 ... P0,L−1 For a,b ∈ {0,1,...,z − 1}, we define a − b as an usual
 
P1,0 P1,1 ... P1,L−1 integer operation modulo z. For a ∈ {0,1,...,z − 1}, we
H = ... ... ... ...  define a−∞ = ∞−a = ∞−∞ = ∞. For vectors v =
 PJ−1,0 PJ−1,1 ... PJ−1,L−1  u(v0=,v(1v,0..−.,uv0L,−v11)−anud1,u..=.,(vuL0−,u11−,.u..L,−u1L)−. 1F)o,rweexadmefipnlee,vw−e
where P is one of a set of z-by-z circulant permutation have (∞,0,1,0,∞)−(1,0,1,∞,0)=(∞,0,0,∞,∞).
i,j
matrices or a z-by-z zero matrix and z is a positive integer.
The matrix H is expanded from a binary base matrix H of C. Quasi-Cyclic LDPC codes of type IEEE802.16e
b
size J-by-L, where m = zJ and n = zL. The base matrix In the standardization IEEE802.16e, quasi-cyclic LDPC
is expanded by replacing each 1 in the base matrix with a codesare chosenas oneofoptimalerror-correctingcodesand
z-by-z circulant permutation matrix, and each 0 with a z-by- six model matrices are written to define quasi-cyclic LDPC
z zero matrix. The circulant permutations used are circular codes.Foreachmodelmatrix,19kindsofthesizeofcireculent
right shifts, and the set of permutation matrices contains matricesare chosen.Totaly,114quasi-cyclicLDPC codesare
the z ×z identity matrix and circular right shifted versions chosen in IEEE802.16e standards. These six model matrices
of the identity matrix. Because each permutation matrix is are designedto satisfy the followingconditions.In this paper,
specifiedbyasinglecircularrightshift,thebinarybasematrix we call a quasi-cyclic LDPC code (a LDPC matrix, a base
information and permutation replacement information can be
matrix, and a model matrix) which satisfies the following
combined into a single compact model matrix H. The model condition of type IEEE802.16e.
matrix H is the same size as the binary base matrix H ,
b Remark 2.3: Theconditionsbelowisnotacharacterization
with each binary entry (i,j) of the base matrix H replaced
b but a generalization of the 6 model matrices in IEEE802.16e.
to create the model matrix H. Each 0 in Hb is replaced by In fact, the set of model matrices which satisfy the condition
a symbol ∞ to denote a z ×z all-zero matrix, and each 1
below contains the 6 model matrices in IEEE.
in H is replaced by a circular shift size p(i,j) ≥ 0. The
b The base matrix H of a quasi-cyclic LDPC matrix is
b
model matrix H can then be directly expanded to H. Denote
partitioned into two sections, where H corresponds to the
b1
a circulant permutation matrix of single shift by I(1), i.e.
systematic bits and H corresponds to the parity-check bits,
b2
1 such that Hb =[(Hb1)J×(L−J)|(Hb2)J×J].
 1  Section Hb2 is further partitioned into two sections, where
I(1)= ... , dhibagisonaalvsetcrtuocrtuwrehowseithwmeiagthritxiseltehmreeen,tsanadt rHowb′2 ih,acsolaumdunalj-
 
 1  equal to 1 for i=j, 1 for i=j+1, and 0 elsewhere:
 
 1 
h (0) 1
b
and I(b) = I(1)b. A circulant permutation Pi,j in H is  hb(1) 1 1 
1I0(pEw(xiai,tmjh)p)thl.ee2f.o2l:lowLeintgHfobrema:parity-checkmatrix of size 6-by- Hb2 =[hb|Hb′2]= ...... 1 ...... 1 .
 
0 0 1 0 0 1 1 0 0 0  .. 
 . 1 1 
 0 0 0 1 1 0 0 1 0 0   
 h (J −1) 1 
b
 1 0 1 0 1 0 1 0 1 0 
H = .
 0 1 0 1 0 1 0 1 0 1  The base matrix has hb(0) = 1,hb(J −1) = 1, and a third
 
 0 1 1 0 0 1 0 0 1 0  value hb(j) = 1 with some 0 < j < J −1. In particular,
 
 1 0 0 1 1 0 0 0 0 1  the non-zero sub-matrices are circularly right shifted by a
particular circular shift value. Each 1 in H′ is assigned a
Then H defines a quasi-cyclic LDPC code with circulant b2
shiftsizeof0,andisreplacedbyaz×z identitymatrixwhen
matrices of size 2. The base matrix H associated to H is
b expandingto H. Thetwo locatedatthe topandthebottomof
a matrix of size 3-by-5:
h areassignedequalshiftsizes, andthe third1 inthemiddle
b
0 1 1 1 0 of hb is given an unpaired shift size.
 
H = 1 1 1 1 1 . Example 2.4: We pick up two base matrices H and
b b(1/2)
 1 1 1 0 1  Hb(2/3B) written in the standardization IEEE802.16e. The
matrix H defines quasi-cyclic LDPC codes of rate 1/2 withcirculantmatricesofsizey.IfC andDsatisfythetwisted
b(1/2)
and has the following form: condition then C′ and D′ satisfy the twisted condition.
Proof: For any positive integers a,b, if a = b (mod z)
011000001100110000000000
  thena mod b (mod y).Itimpliesthatifavectorvover[z]∞
010001110001011000000000
is multiplicity-eventhen the vector v is multiplicity-evenas a
 000111010001001100000000 
  vector over [y]∞.
 101000001100000110000000 
  Let c (resp. d ) be the ith row of the model matrix H
 001000100110000011000000  i i C
  (resp. H ) of C (resp. D). By Theorem 2.5, c − d is
 000011010001100001100000  D j k
Hb(1/2) = 001100000110000000110000 . multiplicity-even as a vector over [z]∞. Thus cj − dk is
  multiplicity-even as a vector over [y]∞.
 011000100100000000011000 
  Corollary 3.2: Let C and D be quasi-cyclic LDPC codes
 100011010001000000001100 
  with circulant matrices. Let H (resp. H ) be the base
 000001010011000000000110  matrix of C (resp. D). Let Cb′(C()resp. D′) bb(eD)a linear code
 001100001100000000000011 
  with a parity-check matrix H (resp. H ). If C and D
 100001010001100000000001  satisfythetwistedcondition,thbe(Cn)C′andD′b(sDat)isfythetwisted
The matrix Hb(2/3B) definesquasi-cyclicLDPC codesof rate condition, i.e. we have
2/3 and has the following form:
H ×HT =0.
b(C) b(D)
101010101010101011000000 Proof: The base matrix H is a quasi-cyclic LDPC
  b(C)
010101010101010101100000 matrix which has the model matrix same as that of C with
 101010101010101000110000  circulant matrices of size 1. By Proposition 3.1, C′ and D′
 
 010101010101010100011000  satisfy the twisted condition.
H = .
b(2/3B)  101010101010101000001100  Remember that two linear codes satisfy the twisted condi-
 
 010101010101010100000110  tion if and only if these parity-checkmatrices H and H are
  1 2
 101010101010101010000011  orthogonal to each other i.e.
 
 010101010101010110000001 
H ×HT =0.
D. Preliminaries of the twisted condition for Quasi-Cyclic 1 2
LDPC codes Thus we have H ×HT =0.
b(C) b(D)
Recall that a CSS code is defined by two classical linear
IV. NON-EXISTANCE RESULTSFOR APAIR OF QC-LDPC
codes.DenotethetwoclassicalcodesbyC andD.Wedenote
CODES OF TYPEIEEE802.16EWITH RATE1/2 AND2/3B
a quasi-cyclic LDPC matrix, the model matrix, and the base
Thecolumn(resp.row)weightdistributionisdefinedasthe
matrixassociatedwithC byH ,H ,andH ,respectively.
C b(C) C
distribution of Hamming weights of the columns (resp. the
Similarly, we use the notation H , H , and H to denote
D b(D) D
rows)fora givenparity-checkmatrix.The weightdistribution
a quasi-cyclic LDPC matrix, the model matrix, and the base
is one of the important parameters for the design of LDPC
matrix associated with D, respectively.
codes. In fact, the performance of LDPC codes with well-
In [7], a necessary and sufficient condition for QC-LDPC
optimized weight distribution is very close to the asymptotic
codes C and D to satisfy (T) in terms of the model matrices
theoreticalbounds[10].Notethattheweightdistributionofthe
has been obtained. We quote the necessary and sufficient
parity-checkmatrix of quasi-cyclicLDPC code with circulant
condition from [7]. Denote j th rows of the model matrices
matrices is the same as that of its base matrix. There are i
C and D by c and d , respectively. We call a vector v = w
j j
columns(resp.rows)ofHammingweightwinthebasematrix
(v0,v1,...,vL−1) over{0,1,...,z−1}∪{∞} multiplicity-
if and only if there i z columns (resp. rows) of Hamming
even if each symbol except for ∞ appears even times in w
weight w in associated parity-check matrix, where z is the
{v0,v1,...,vL−1}. For example, a vector (0,0,1,1,1,∞,1)
size of the circulant matrices.
is multiplicity-even.
In this section, we discuss our problem, which is a con-
Theorem 2.5 ([7] Prop. 3.1. and Sec. IV(D)): Let C and
struction of a pair of irregular LDPC codes to derive a CSS
D be quasi-cyclic LDPC codes. The codes C and D satisfy
code, under the following conditions:
(T) if and only if c −d is multiplicity even for any row c
j k j
of the model matrix H and any row d of the modelmatrix • Classical code C and D are quasi-cyclic LDPC codes
C k
H . with circulant matrices of size z.
D
• The associated low-density parity-check matrices of C
III. A NECESSARY CONDITION FOR THETWISTED and D have the same “row-weight” distributions.
CONDITION IN TERMS OF THEBASE MATRICES • The weight of columns is more than or equal to 2. This
Proposition 3.1: LetC andD bequasi-cyclicLDPCcodes conditionarisesfromastandarddecodingmethod,which
with circulant matrices of size z. Let y be a positive integer is called a sum-product decoding. If one of a column-
such that y divides z. Let C′ (resp. D′) be a quasi-cyclic weightislessthan2,thenthesum-productdecodingdoes
LDPC code with the model matrix same as C (resp. D) and not work well [12].
Inthispaper,denotetheconditionaboveby(I).Thecondition B. Construction Impossibility on D under H :=H
b(C) b(1/2)
(I) does not impose the limitation on the base matrices. Let C be a quasi-cyclic LDPC code with its base matrix
In subsection §IV-A, we discuss the impossibility on con- H :=H .
b(C) b(1/2)
structionofD underHb(C) :=Hb(2/3B).Similarly,insubsec- By the twisted condition, for the base matrix Hb(D) of a
tion §IV-B, the impossibility on construction under Hb(C) := quasi-cyclicLDPCcodeD,thefollowingequationisrequired:
H is discussed.
b(1/2) H ×HT =0.
b(C) b(D)
A. ConstructionImpossibilityonD underHb(C) :=Hb(2/3B) It implies that any row x∈F224 of Hb(D) satisfies:
Let C be a quasi-cyclic LDPC code with the base matrix H x=0.
b(C)
H :=H .
b(C) b(2/3B)
The weight of each row of H is 6 or 7. By the definition
It is required that the base matrix H of D satisfies: b(D)
b(D) of a quasi-cyclic LDPC code of type IEEE802.16e, two non-
H ×HT =0. zeroentriesofanyrowofHb(D),exceptforthebelow,appear
b(C) b(D) inadjacentpositiontoeachother.Theweightoftheright-side
It implies that any row x∈F224 of Hb(D) satisfies: 11 bits of a row of Hb(D) is one or two and we denote the
weight of the right-side by wt|13−.
Hb(C)x=0. Define three sets X1,X2 and X3:
X :={x∈F24|wt(x)=6 or 7},
Define a matrix M as follows: 1 2
X :={x∈F24|x =x =1 for some 14 ≤i≤23},
11111111 2 2 i i+1
 01111111  X3 :={x∈F224|wt|13−(x)=1 or 2}.
 00111111 
  Forsimplicity,wedenoteX :=X ∩X ∩X .Theneachrow
 00011111  1 2 3
M := . of H belongs to the set X.
 00001111  b(D)
  We can determine the set X by using a personal com-
 00000111 
  puter. In fact, we verify X consists of six elements. The
 00000011 
  right-side 12-bits of each element of X have the form
 11111110 
(0,0,0,0,0,0,0,0,1,1,0,0).On the otherhand,elevenkinds
Then, for a vector x∈F24, we have oftheright-side12-bitsarerequiredtosatisfy(I).Itshowsthat
2 it is impossible to construct H which satisfies (I) under
b(D)
Hb(C)x=0 ⇐⇒ MHb(C)x=0. the assumption Hb(C) :=Hb(1/2).
By the result, we immediately have the
By an easy calculation, we obtain Corollary 4.2: LetCbeoneof19quasi-cyclicLDPCcodes
which has the “base matrix” written as rate 2/3B codes in
000000000000000010000000 IEEE802.16e. Then there is no quasi-cyclic LDPC code D
 
101010101010101001000000 which satisfies (I).
 111111111111111100100000  Proof: C is a particular example which satisfies our
 
 010101010101010100010000  condition.
MH = .
b(C)  000000000000000000001000 
 101010101010101000000100  V. CONCLUSION
 
 111111111111111100000010  Inthispaper,wediscusstheconditionstoconstructapairof
 
 010101010101010100000001  quasi-cyclic LDPC codes, in particular of type IEEE802.16e,
to derive a CSS code. The key of our discussion is to
By considering the first and the fifth rows, we obtain a analyzethebasematricesofquasi-cyclicLDPCcodes.Byour
necessary condition x17 = x21 = 0 for Hb(C)x = 0, where research,wefindtheimpossibilitytoconstructapairofquasi-
x=(x1,x2,...,x24). cyclic LDPC codes to derive a CSS code under conditions
Thusthe weightof 17th columnand 21st columnof Hb(D) that (I) holds and one of the base matrices is the same as an
mustbe0,inparticularislessthan2.Thereforeitisimpossible IEEE802.16e LDPC base matrix with rate 1/2 or 2/3B. Note
toconstructD whichsatisfiesthecondition(I)underHb(C) = that our results are more general. The 32 quasi-cyclic LDPC
Hb(2/3B). By the result, we immediately have the codes, which have the model matrice written as rate 1/2 and
Corollary 4.1: LetC beoneof19quasi-cyclicLDPCcodes rate2/3BinIEEE802.16e,aretypicalexamplesofourresults.
which has the “base matrix” written as rate 1/2 codes in Weshouldrememberthatsixbasematrices,totally96quasi-
IEEE802.16e. Then there is no quasi-cyclic LDPC code D cyclic LDPC codes, are discribed in the standardization form
which satisfies (I). of IEEE802.16e.Hope has been left for the other four LDPC
Proof: C is a particular example which satisfies our codes, totally 76 quasi-cyclic LDPC codes. We leave these
condition. posibilities as open problems.
ACKNOWLEDGEMENT
This research was partially supported by Grants-in-Aid for
Young Scientists (B), 18700017, 2006.
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