Quantum synchronizable codes from finite geometries
aa r X i v : . [ qu a n t - ph ] S e p IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. X, NO. XX, MONTH YEAR 1
Quantum Synchronizable Codes from FiniteGeometries
Yuichiro Fujiwara,
Member, IEEE and Peter Vandendriessche,
Member, IEEE
Abstract —Quantum synchronizable error-correcting codes arespecial quantum error-correcting codes that are designed to cor-rect both the effect of quantum noise on qubits and misalignmentin block synchronization. It is known that in principle such acode can be constructed through a combination of a classicallinear code and its subcode if the two are both cyclic and dual-containing. However, finding such classical codes that lead topromising quantum synchronizable error-correcting codes is nota trivial task. In fact, although there are two families of classicalcodes that are proved to produce quantum synchronizable codeswith good minimum distances and highest possible toleranceagainst misalignment, their code lengths have been restrictedto primes and Mersenne numbers. In this paper, examining theincidence vectors of projective spaces over the finite fields ofcharacteristic , we give quantum synchronizable codes fromcyclic codes whose lengths are not primes or Mersenne numbers.These projective geometric codes achieve good performance inquantum error correction and possess the best possible abilityto recover synchronization, thereby enriching the variety ofgood quantum synchronizable codes. We also extend the currentknowledge of cyclic codes in classical coding theory by explicitlygiving generator polynomials of the finite geometric codes andcompletely characterizing the minimum weight nonzero code-words. In addition to the codes based on projective spaces, wecarry out a similar analysis on the well-known cyclic codes fromEuclidean spaces that are known to be majority logic decodableand determine their exact minimum distances. Index Terms —Quantum error correction, synchronizable code,cyclic code, finite geometry.
I. I
NTRODUCTION Q UANTUM error correction is a crucial element ofquantum information science that plays a key role inrealizing quantum information processing in a noisy environ-ment. Active quantum error correction is an important andextensively studied method for suppressing quantum noise,where one extracts the information about what types of errorsoccurred on which qubits through syndrome extraction withoutdisturbing the quantum information carried by qubits. Oncethis information is obtained, the effect of quantum noise maybe nullified by applying appropriate quantum operations onaffected qubits.
This work was supported by JSPS (Y.F.) and FWO (P.V.). The second authoris support by a PhD fellowship of the Research Foundation - Flanders (FWO).Y. Fujiwara is with the Division of Physics, Mathematics and Astronomy,California Institute of Technology, MC 253-37, Pasadena, CA 91125 USA(email: [email protected]).P. Vandendriessche is with the Department of Mathematics, Ghent Univer-sity, Krijgslaan 281 - S22, 9000 Ghent, Belgium (email: [email protected]).Copyright c (cid:13)
In the context of quantum error correction, quantum noiseis most typically described by operators that act on qubits.The most general error model of this kind is the linearcombinations of the Pauli operators I , X , Y , and Z actingon each qubit [1]. From this point of view, quantum error-correcting codes are schemes that allow for recovering theoriginal quantum state when unintended operators may act onsome qubits.This type of typical error model may be considered aquantum version of additive noise , which is among the mostimportant and well-studied kinds of error models in infor-mation theory. However, it is not the only error model ofimportance.An example of errors that do not fall into the categoryof additive noise but are crucial in information theory issynchronization errors. The simplest type of synchronizationerror is misalignment with respect to the frame structure ofa data stream. To describe misalignment in the context ofquantum information, assume that we have three qubits q , q , q and encode each of them by the perfect -qubit code [2],[3]. Then the quantum information we have can be expressedby a sequence of fifteen qubits, where each -qubit state | ψ i i , i = 0 , , , represents one logical qubit of quantuminformation that corresponds to the original qubit q i . In orderto correctly process quantum information, we need to knowthe exact location of the boundary of each -qubit block inthe -qubit state | ψ i | ψ i | ψ i . For instance, if misalignmentoccurs by two qubits to the left when handling the stream offifteen qubits, our quantum error correction device trying tocorrect errors on | ψ i will apply the quantum operation on thewrong set of five qubits, two of which come from | ψ i andthree of which belong to | ψ i .In classical coding theory, error-correcting codes that cancorrect both additive noise and misalignment in block synchro-nization are called synchronizable error-correcting codes [4].This paper studies the quantum analogue of such active errorcorrection schemes that allow for extracting the informationabout the magnitude and direction of misalignment throughmeasurement while simultaneously identifying the types andpositions of standard quantum errors that may have occurredon qubits.Formally speaking, we call a coding scheme a quantumsynchronizable ( a l , a r ) - [[ n, k ]] code if it encodes k logicalqubits into n physical qubits and corrects misalignment byup to a l qubits to the left and up to a r qubits to the right.In order to suppress as diverse quantum noise as possible, wewould also like our quantum synchronizable codes to be ableto correct linear combinations of I , X , Z , and Y that act on IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. X, NO. XX, MONTH YEAR physical qubits.While it may appear quite difficult to devise an efficientand error-tolerant scheme that also corrects misalignment inthe context of quantum information, it has been proved that inprinciple a known quantum error correction technique can beextended to the case of block synchronization recovery. Thefirst examples of quantum synchronizable codes with standardquantum error correction capabilities were presented in [5],where a general framework for code construction as well astheir encoding and decoding methods were developed. Thisframework was subsequently improved by a more extensiveuse of finite algebra in [6]. In both cases, classical error-correcting codes with special algebraic properties are exploitedin a way similar to the well-known Calderbank-Shor-Steane(CSS) construction [7]–[9].However, these first theoretical steps towards solving theproblem of block synchronization for quantum informationleft many challenges as well. One of the main hurdles in thetheoretical study of quantum synchronizable codes is that it isquite difficult to find suitable classical error-correcting codesbecause the required algebraic constraints are very severe anddifficult to analyze. In fact, there are only two known classesof classical error-correcting codes that are proved to possessthe required properties while achieving good quantum errorcorrection capabilities and high tolerance against misalignmentat the same time.Particularly constrained is the variety of available code pa-rameters. For instance, the lengths of the encoded informationblocks of the known quantum synchronizable codes that havehighest possible tolerance against misalignment are all primesor of the form m − , that is, Mersenne numbers.The primary purpose of this paper is to enrich the spectrumof quantum synchronizable codes by giving an infinite familyin which the encoded block lengths can be neither primes norMersenne numbers. We prove that these codes have the highestpossible tolerance against misalignment and are capable ofcorrecting the effect of a substantial level of standard quantumnoise. To this end, we use the theory of finite geometries andintroduce a class of classical error-correcting codes suitablefor use as the ingredients of quantum synchronizable codes.As a by-product of our analysis, we also extend the currentknowledge of classical coding theory. We exploit one of themost important classes of classical error-correcting codes,namely cyclic codes [10]. While cyclic codes are of practicaland theoretical importance, their true minimum distances arenotoriously difficult to compute. In the study that follows, weshow various properties of cyclic codes based on two types offinite geometries, that is, projective geometry and Euclideangeometry. For the case of projective geometry, not only do weprove the exact minimum distances, but we also completelycharacterize the nonzero codewords of minimum weight. Asimilar analysis is carried out for a well-known class of cycliccodes from Euclidean geometry which have extensively beenstudied both in classical coding theory and in quantum codingtheory. We determine the exact minimum distances of theEuclidean cyclic codes and also prove that they are capable ofrecovering synchronization from severe misalignment if usedas quantum synchronizable codes. In the next section we briefly review the improved frame-work of quantum synchronizable codes given in [6] fromthe viewpoint of what kind of classical error-correcting codeis required. Section III examines special structures of finitegeometries and shows that they form suitable classical error-correcting codes for our purpose. Concluding remarks aregiven in Section IV.II. G ENERAL CONSTRUCTION
This section describes the properties of classical error-correcting codes required for constructing quantum synchro-nizable error-correcting codes. The proofs of the basic facts incoding theory we use in this paper can be found in a standardtextbook such as [11]. For mathematical details of the problemof block synchronization for quantum information, we referthe reader to [6], [12].A binary linear [ n, k, d ] code of length n , dimension k ,and minimum distance d is a k -dimensional subspace L ofthe n -dimensional vector space F n over the finite field F of order such that min { wt ( v ) | v ∈ L \ { }} = d , wherewt ( v ) is the number of nonzero coordinates of v . Let C and D be two binary linear codes of the same length. D is C - containing if C ⊆ D . It is dual-containing if it contains itsdual D ⊥ = (cid:8) d ⊥ ∈ F n | d · d ⊥ = , d ∈ D (cid:9) . In what follows,we always assume that classical codes are over F and omitthe term binary.A cyclic [ n, k, d ] code C is a linear [ n, k, d ] code inwhich every cyclic shift of every codeword c ∈ C is alsoa codeword, that is, for any c = ( c , . . . , c n − ) ∈ C , wehave ( c , . . . , c n − , c ) ∈ C . It is known that, by regardingeach codeword as the coefficient vector of a polynomial in F [ x ] , a cyclic code of length n can be seen as a principalideal in the ring F [ x ] / ( x n − generated by the uniquemonic nonzero polynomial g ( x ) of minimum degree in thecode which divides x n − . When a cyclic code is of length n and dimension k , the set of codewords can be written as C = { i ( x ) g ( x ) | deg( i ( x )) < k } , where the degree deg( g ( x )) of the generator polynomial is n − k .The improved framework for constructing quantum synchro-nizable codes involves an algebraic notion in polynomial rings.Let f ( x ) ∈ F [ x ] be a polynomial over F such that f (0) = 1 .The cardinality ord( f ( x )) = |{ x a (mod f ( x )) | a ∈ N }| iscalled the order of the polynomial f ( x ) , where N is the setof positive integers. This cardinality is also known as the period or exponent of f ( x ) in the literature. The followingis the improved general construction stated in the form of amathematical theorem. Theorem 2.1 ([6]):
Let C be a dual-containing cyclic [ n, k , d ] code with generator polynomial h ( x ) and D a C -containing cyclic [ n, k , d ] code with generator polyno-mial g ( x ) . Define polynomial f ( x ) of degree k − k tobe the quotient of h ( x ) = f ( x ) g ( x ) divided by g ( x ) over F [ x ] / ( x n − . For every pair a l , a r of nonnegative integerssuch that a l + a r < ord( f ( x )) there exists a quantumsynchronizable ( a l , a r ) - [[ n + a l + a r , k − n ]] code thatcorrects at least up to (cid:4) d − (cid:5) phase errors and at least upto (cid:4) d − (cid:5) bit errors. UJIWARA AND VANDENDRIESSCHE: QUANTUM SYNCHRONIZABLE CODES FROM FINITE GEOMETRIES 3
For the proof of the theorem above and the procedures forencoding and decoding, the reader is referred to [5], [6].From the viewpoint of searching for suitable classical error-correcting codes, an important fact to note is that if a linearcode C is dual-containing, a C -containing linear code is alsodual-containing. Hence, what Theorem 2.1 actually requires isa pair of dual-containing cyclic codes, one of which is con-tained in the other and both of which guarantee large minimumdistances. In addition to being nested, dual-containing, cyclic,and of large minimum distance, it is desirable for the pair oflinear codes to lead to as large ord( f ( x )) as possible in orderto tolerate the widest range of misalignment. Note that forany pair of cyclic codes of length n , the corresponding valueof ord( f ( x )) is at most n . This is because f ( x ) is a divisorof the generator polynomial of the smaller cyclic code. Inother words, f ( x ) also divides x n − , so that x a = x a + n (mod f ( x )) for any integer a .The known quantum synchronizable codes employ well-known classes of cyclic codes called narrow-sense Bose-Chaudhuri-Hocquenghem (BCH) codes and punctured Reed-Muller codes . Their precise definitions, parameters, and otherimportant facts in the context of quantum block synchroniza-tion can be found in [6] and references therein. The notablepoint is that these cyclic codes have substantial minimumdistances while being both dual-containing and nested if theirdimensions are large enough. It can be shown that the cor-responding orders ord( f ( x )) often takes the maximum value.However, the lengths of the codes of the former class whichhas been studied for synchronization recovery to a substantialdepth are all of the form m − or primes. The latter classonly contains codes of length that is simultaneously prime andof the form m − . The goal of the next section is to provethat these are not the only suitable error-correcting codes bygiving explicit examples whose lengths have not be realizedby the previously known families.III. S UBSPACES OF FINITE GEOMETRIES AND QUANTUMSYNCHRONIZABLE CODES
This section examines properties of finite geometries andprovide a family of quantum synchronizable codes. The proofsof the basic facts and notions regarding finite geometries weneed can be found in [13].We divide this section into two subsections. Section III-Astudies codes based on projective geometry. Codes based onEuclidean geometry are examined in Section III-B. In bothcases, the lengths, dimensions, and minimum distances ofour classical error-correcting codes are precisely and theoret-ically determined. We also prove that the maximum tolerablemagnitudes ord( f ( x )) of misalignment of the correspondingquantum synchronizable codes are always the same as theircode lengths, which are the highest possible values. A. Projective geometry codes
Let m , h , and t be positive integers such that t ≤ m − . The projective geometry PG ( m, h ) of dimension m over F h is afinite geometry whose points and t - dimensional subspaces arethe -dimensional vector subspaces and the ( t +1) -dimensional vector subspaces of the ( m + 1) -dimensional vector space F m +12 h respectively. Because the points are the -dimensionalvector subspaces, the set P of points in PG ( m, h ) is ofcardinality h ( m +1) − h − .Take a t -dimensional subspace π of PG ( m, h ) . The in-cidence vector χ π of π is the binary h ( m +1) − h − -dimensionalvector such that the coordinates are indexed by the points andsuch that each entry is if π contains the corresponding pointand otherwise. Let B be the set of t -dimensional subspacesof PG ( m, h ) . It is known that there exists a collineation σ such that the group h σ i of order h ( m +1) − h − acts regularly onthe points in PG ( m, h ) [14], which means that the incidencerelation in the set system ( P, B ) is invariant under the action ofthe cyclic group h σ i . Therefore, by regarding points as the ele-ments of the cyclic group and indexing the coordinates of eachincidence vector accordingly by σ , σ , σ, , . . . , σ h ( m +1) − h − in the natural order, for any incidence vector χ π = ( x , . . . , x h ( m +1) − h − − ) of a t -dimensional subspace π ∈ B , its cyclic shift ( x , . . . , x h ( m +1) − h − − , x ) is also the incidence vector of some t -dimensional subspace.Hence, the vector space P m,t, h = h χ π | π ∈ Bi spanned bythe incidence vectors of t -dimensional subspaces in PG ( m, h ) can be seen as a cyclic code of length h ( m +1) − h − . We assumethat the coordinates are indexed by the points of PG ( m, h ) in this cyclic way throughout this subsection.The complement χ π of an incidence vector χ π is thevector χ π = χ π + , where is the all-one vector. In otherwords, χ π is obtained by flipping all ’s and ’s in χ π . Let C m,t, h = h χ π | π ∈ Bi ⊥ be the dual of the vector spacespanned by the set of complements of the incidence vectors of t -dimensional subspaces of PG ( m, h ) . We prove that C m,t, h satisfies the required conditions for use as ingredients of quan-tum synchronizable codes while achieving good quantum errorcorrection capabilities and high tolerance against misalignmentif t is in a suitable range with respect to the size of m . Morespecifically, we show the following theorem. Theorem 3.1:
Let m , h , and t be positive integerssuch that m +12 ≤ t ≤ m − . For every pair a l , a r of nonnegative integers such that a l + a r < h ( m +1) − h − there exists a quantum synchronizable ( a l , a r ) - hh h ( m +1) − h − + a l + a r , h ( m +1) − h − − P m,t, h + 2 ii code that corrects at least up to h ( m − t +1) − − h − h − phaseerrors and at least up to h ( m − t ) − − h − h − bit errors.As we have seen in the previous section, Theorem 2.1requires classical error-correcting codes to simultaneouslysatisfy various properties. The fact that C m,t, h = h χ π | π ∈ Bi ⊥ is a cyclic code follows directly from the fact that P m,t, h = h χ π | π ∈ Bi is cyclic as a linear code. Proposition 3.2: C m,t, h is cyclic as a linear code.To form a quantum synchronizable code, a cyclic code mustbe dual-containing. Lemma 3.3: If m +12 ≤ t ≤ m − , then C ⊥ m,t, h ⊆ C m,t, h . IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. X, NO. XX, MONTH YEAR
Proof:
It suffices to show that for any pair π , π ∈ B of t -dimensional subspaces of PG ( m, h ) , the correspondingincidence vectors χ π , χ π ∈ C ⊥ m,t, h are orthogonal to eachother. Note that χ π and χ π are orthogonal to each other ifand only if the cardinality | π ∩ π | is even. Note also that | π ∩ π | = 2 h ( m +1) − h − − | π ∪ π | = 2 h ( m +1) − h − − | π | − | π | + | π ∩ π | . If t ≥ m +12 , the intersection π ∩ π is a nonempty subspaceof PG ( m, h ) . Because any nonempty subspace of PG ( m, h ) contains an odd number of points, the four terms h ( m +1) − h − , | π | , | π | , and | π ∩ π | are all odd. Thus, their sum, whoseparity is the same as that of | π ∩ π | , is indeed even.The following lemma shows that C m,t, h has a suitablenested property. Lemma 3.4: If ≤ t ≤ m − , then C m,t − , h ⊂ C m,t, h .Note that recursively applying the above lemma shows that C m,t, h contains C m,t ′ , h for all t ′ ≤ t . To prove Lemma 3.4,we use two simple facts about our cyclic codes. Proposition 3.5: P m,t, h = hC ⊥ m,t, h , i , where the h ( m +1) − h − -dimensional all-one vector
6∈ C ⊥ m,t, h Proof:
Because all generators χ π of C ⊥ m,t, h are of evenweight, all codewords of C ⊥ m,t, h are also of even weight.Since the length h ( m +1) − h − of this cyclic code is odd, theall-one vector is not a codeword. Recall that P m,t, h is thevector space spanned by the incidence vectors of t -dimensionalsubspaces in PG ( m, h ) . Because the number of t -dimensionalsubspaces that contain a given point in PG ( m, h ) is alwaysodd, we have X π ∈B χ π = , which implies that ∈ P m,t, h . Because χ π = χ π + , itfollows that C ⊥ m,t, h ⊂ P m,t, h . Thus, we have hC ⊥ m,t, h , i ⊆P m,t, h . Now the fact that χ π = χ π + is equivalent to therelation that χ π = χ π + , which implies that P m,t, h ⊆hC ⊥ m,t, h , i . Thus, P m,t, h = hC ⊥ m,t, h , i as desired. Proposition 3.6: C m,t, h = hP ⊥ m,t, h , i , where ⊥ m,t, h . Proof:
Because the generators of P m,t, h are all of oddweight, the inner product between and any of the generatorsis nonzero, which implies that
6∈ P ⊥ m,t, h . By the same token,because the generators of C ⊥ m,t, h are of even weight, we have ∈ C m,t, h . By Proposition 3.5, C ⊥ m,t, h ⊂ P m,t, h , whichimplies that P ⊥ m,t, h ⊂ C m,t, h . Again by Proposition 3.5, thedimensions of P m,t, h and C ⊥ m,t, h satisfy the equation that dim P m,t, h = dim C ⊥ m,t, h +1 . Hence, C m,t, h = hP ⊥ m,t, h , i as desired. Proof of Lemma 3.4:
Assume that ≤ t ≤ m − . Take a t -dimensional subspace π of PG ( m, h ) and define Π to be theset of ( t − -dimensional subspaces that are contained in π .The set Π contains exactly ht − h − ( t − -subspaces containinga given point p ∈ π . Since this number is odd, we have χ π = X π ′ ∈ Π χ π ′ . Thus, P m,t, h ⊂ P m,t − , h , which is equivalent to P ⊥ m,t − , h ⊂ P ⊥ m,t, h . Hence, by Proposition 3.6, we have C m,t − , h ⊂ C m,t, h . The proof is complete.Proposition 3.2 and Lemmas 3.3 and 3.4 together provethat for any pair t , t ′ of positive integers such that ≤ t ′ < t ≤ m − , the pair C m,t, h , C m,t ′ , h of linear codessatisfy the three conditions of cyclicity, duality, and nestednessrequired to construct a quantum synchronizable code throughTheorem 2.1. Because we already know that their length is h ( m +1) − h − , the remaining task is to determine their dimensions,minimum distances, and the maximum tolerable magnitude ofmisalignment.The dimension of C m,t, h is determined by that of P m,t, h . Lemma 3.7:
For positive integers m , h , t such that t ≤ m − , dim C m,t, h = 2 h ( m +1) − h − − dim P m,t, h + 1 . Proof:
By Proposition 3.6, we have dim C m,t, h = dim P ⊥ m,t, h + 1= min {| P | , |B|} − dim P m,t, h + 1 , where P is the set of points and B is the set of t -dimensionalsubspaces of PG ( m, h ) . The number h ( m +1) − h − of pointsis always smaller than or equal to that of t -dimensionalsubspaces.The following well-known formula gives the exact value of dim P m,t, h , allowing for calculating the dimensions of ourcyclic codes. Theorem 3.8 ([15]):
For positive integers m , h , t such that t ≤ m − , dim P m,t, h = X ( s ,s ,...,s h ) h − Y j =0 j sj +1 − sj k X i =0 ( − i (cid:18) m + 1 i (cid:19) × (cid:18) m + 2 s j +1 − s j − im (cid:19) , where the first summation is taken over all ( s , s , . . . , s h ) with s h = s ; t + 1 ≤ s j ≤ m + 1 for all j with ≤ j < h ,and ≤ s j +1 − s j ≤ m + 1 for all j with ≤ j < h .To examine the minimum distance of C m,t, h , we employthe following two theorems. Theorem 3.9 ([16]):
The minimum distance of P ⊥ m,t, h is (2 h + 2)2 h ( m − t − . Theorem 3.10 ([17]):
The codewords of P ⊥ m,t, h that havethe largest number of nonzero entries are of weight h ( m +1) (1 − − ht )2 h − .We are now able to give the complete profile of theparameters of C m,t, h as a linear code. Theorem 3.11: C m,t, h is a linear [ h ( m +1) − h − , h ( m +1) − h − − dim P m,t, h + 1 , h ( m − t +1) − h − ] code. Proof:
It suffices to prove that the minimum distanceis exactly h ( m − t +1) − h − . By Proposition 3.6, a codeword of C m,t, h is either contained in P ⊥ m,t, h or of the form c + forsome c ∈ P ⊥ m,t, h . By Theorem 3.9, among the codewords of UJIWARA AND VANDENDRIESSCHE: QUANTUM SYNCHRONIZABLE CODES FROM FINITE GEOMETRIES 5 the former kind, the ones with the smallest number of nonzeroentries are of weight exactly (2 h + 2)2 h ( m − t − . Among thecodewords of the latter kind, the ones with the smallest numberof nonzero entries are those obtained as the complements ofthe codewords of P ⊥ m,t, h that have the largest number ofnonzero entries. Hence, by Theorem 3.10, the codewords of thelatter kind that have the smallest number of nonzero entries areof weight exactly h ( m +1) − h − − h ( m +1) (1 − − ht )2 h − = h ( m − t +1) − h − .For all positive h and positive t < m , we have h ( m − t +1) − h − < (2 h + 2)2 h ( m − t − .An interesting observation is that the maximum weightcodewords in the dual P ⊥ m,t, h are the incidence vector ofthe complement of ( m − t ) -spaces in PG ( m, h ) (see [17,Theorem 3.1]). Thus, the proof above actually shows that theminimum weight nonzero codewords in C m,t, h are exactly theincidence vectors of ( m − t ) -spaces in PG ( m, h ) , giving acomplete picture of how the nonzero codewords of minimumweight are formed and how many there are.As shown in Theorem 3.11, C m,t, h has a nontrivially largeminimum distance and dimension as a cyclic code. Table I liststhe parameters for some m , h , and t . When h = 1 , the lengthof C m,t, h is not a Mersenne number and can generally be acomposite number, which has not been realized by previouslyknown cyclic codes that lead to quantum synchronizable codeswith excellent synchronization recovery capabilities. It shouldbe noted that if h = 1 , primitive BCH codes of the samelength are in general as good or better in terms of dimensionand minimum distance.The final criterion for being ideal ingredients of quantumsynchronizable codes is to be able to provide high toleranceagainst synchronization errors. We prove that any pair of cycliccodes in the nested chain C m, ⌈ m +12 ⌉ , h ⊂ C m, ⌈ m +12 ⌉ +1 , h ⊂· · · ⊂ C m,m − , h attain the upper bound on the synchroniza-tion recovery capabilities.To investigate the tolerable magnitude of misalignment, wefirst determine the generator polynomial of C m,t, h and thenapply theorems in finite algebra to explicitly spell out the exactvalues of ord( f ( x )) in Theorem 2.1 for the case when thecyclic codes are chosen from our nested chain.To this end, we use the weight w h ( a ) of the h -aryexpansion of a positive integer a , that is, w h ( a ) = X i a i , where addition is performed over Z and a = X i ∈ N ∪{ } a i hj with ≤ a i ≤ h − . The following theorem gives the explicitform of the generator polynomial of C m,t, h . Theorem 3.12:
Let α be a primitive element in F h ( m +1) and β = α h − . The generator polynomial g ( x ) of C m,t, h is g ( x ) = Y j ∈ I m,t,h ( x − β j ) , TABLE IS
AMPLE PARAMETERS OF C m,t, h FROM PG ( m, h ) . m h t Length Dimension Minimum distance4 1 2 31 16 74 1 3 31 26 34 2 2 341 196 214 2 3 341 316 54 3 2 4681 3106 734 3 3 4681 4556 95 1 3 63 42 75 1 4 63 57 35 2 3 1365 1064 215 2 4 1365 1329 55 3 3 37449 32598 735 3 4 37449 37233 96 1 3 127 64 156 1 4 127 99 76 1 5 127 120 36 2 3 5461 3186 856 2 4 5461 4901 216 2 5 5461 5412 57 1 4 255 163 157 1 5 255 219 77 1 6 255 247 37 2 4 21845 16629 857 2 5 21845 20885 217 2 6 21845 21781 58 1 4 511 256 318 1 5 511 382 158 1 6 511 466 78 1 7 511 502 38 2 4 87381 51396 3418 2 5 87381 76512 858 2 6 87381 85836 218 2 7 87381 87300 5 where I m,t,h = (cid:26) a ∈ N (cid:12)(cid:12)(cid:12)(cid:12) a ≤ h ( m +1) − h − , max ≤ i ≤ h w h ( a (2 h − i ) ≤ ( m − t )(2 h − (cid:27) . Proof:
It is known that P m,t, h is the subfield subcodein F of a punctured generalized Reed-Muller codes [18]. Thegenerator polynomial h ( x ) of its dual P ⊥ m,t, h is h ( x ) = ( x − Y j ∈ I m,t,h ( x − β j ) (see [19, Theorem 13.9.2] ). It suffices to show that h ( x ) =( x − g ( x ) . By Lemma 3.6, P ⊥ m,t, h ⊂ C m,t, h and To avoid confusion in notation, “ m ” in Theorem 13.9.2 in [19] correspondsto “ m + 1 ” in this paper while “ r ” and “ s ” there are “ t ” and “ h ” hererespectively. Note also that the current edition of the textbook containstypographical errors in the statement of Theorem 13.9.2, so that “ < j . . . ”and “ < max . . . ” should read “ ≤ j . . . ” and “ ≤ max . . . ”respectively. IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. X, NO. XX, MONTH YEAR dim C m,t, h = dim P ⊥ m,t, h + 1 . Thus, the generator polyno-mial g ( x ) of C m,t, h is a divisor of h ( x ) , where the quotientis a polynomial of degree over F . Since h (0) = 1 ,the polynomial x is not a factor of h ( x ) . Hence, we have h ( x ) = ( x − g ( x ) as desired.To show that C m,t, h gives the highest possible toleranceagainst misalignment, we use the following tools in finitealgebra. Proposition 3.13:
Let f ( x ) = Q i f i ( x ) be a polynomialover F , where f i ( x ) are all nonzero and pairwise relativelyprime in F [ x ] . Then ord( f ( x )) = lcm i { ord( f i ( x )) } . Proposition 3.14:
Let q be a prime power and α a nonzeroelement of the extension field F q e of F q for a positive integer e . Define f ( x ) ∈ F q [ x ] to be the minimial polynomial of α over F q . The order ord( f ( x )) is equal to the order of α in themultiplicative group F ∗ q e .For the proofs of these propositions, see [20, Theorems 3.9and 3.33].We now prove that a pair C m,t, h , C m,t ′ , h of cyclic codesachieves the trivial upper bound on the maximum tolerablemagnitude of misalignment when used as ingredients in The-orem 2.1. Lemma 3.15:
Let g ( x ) and h ( x ) be the generator polynomi-als of C m,t, h and C m,t − i, h for a positive integer i ≤ t − re-spectively. Define f ( x ) to be the quotient of h ( x ) = f ( x ) g ( x ) divided by g ( x ) . Then ord( f ( x )) = h ( m +1) − h − . Proof:
Let α be a primitive element in F h ( m +1) and β = α h − . By Theorem 3.12, we have f ( x ) = Y j ∈ I m,t − i,h \ I m,t,h ( x − β j ) . Note that every irreducible factor of f ( x ) over F [ x ] is ofmultiplicity . We consider two special factors of f ( x ) . Let j = h ( m − t ) − h − and j = h ( m − t +1) − h − − . It is straightforwardto see that max ≤ i ≤ h w h ( j (2 h − i ) = max ≤ i ≤ h w h ( j (2 h − i )= ( m − t )(2 h − . Hence, the minimal polynomials M β j ( x ) , M β j ( x ) of β j and β j are nonzero factors of f ( x ) . If ord( M β j ( x )) =ord( M β j ( x )) , because j and j are relatively prime, we have ord( f ( x )) ≥ ord( M β j ( x ))= 2 h ( m +1) − (cid:0) h − , h ( m +1) − (cid:1) = 2 h ( m +1) − h − . If ord( M β j ( x )) = ord( M β j ( x )) , because M β j ( x ) and M β j ( x ) are minimal polynomials, the two are relativelyprime. Hence, by Propositions 3.13 and 3.14 and the fact that j and j are relatively prime, we have ord( f ( x )) ≥ lcm (cid:0) ord (cid:0) M β j ( x ) (cid:1) , ord (cid:0) M β j ( x ) (cid:1)(cid:1) = lcm h ( m +1) − (cid:0) j (2 h − , h ( m +1) − (cid:1) , h ( m +1) − (cid:0) j (2 h − , h ( m +1) − (cid:1) ! = 2 h ( m +1) − h − . Since the order of a factor of the generator polynomial of acyclic code is at most the length of the code, in each case wehave ord( f ( x )) = h ( m +1) − h − .We conclude this subsection with the proof of our maintheorem on quantum synchronizable codes from projectivegeometry PG ( m, h ) . Proof of Theorem 3.1:
Take C m,t, h and C m,t +1 , h . ByProposition 3.2, the two are both cyclic codes. Lemma 3.3ensures that they are dual-containing. Lemma 3.4 guaranteesthat C m,t +1 , h is C m,t, h -containing. Applying Theorem 2.1with Theorem 3.11 and Lemma 3.15 proves the assertion. B. Euclidean geometry codes
This subsection investigates a different kind of finite geom-etry. As in the case of projective geometry, let m , h , and t be positive integers such that t ≤ m − . The affine geometry AG ( m, h ) of dimension m over F h is defined as a finitegeometry in which the points are the vectors in F m h and the t - dimensional subspaces are the t -dimensional vector subspacesof F m h and their cosets. Take an arbitary point in AG ( m, h ) and call it the origin . The Euclidean geometry EG ( m, h ) of dimension m over F h is the finite geometry obtainedby deleting from AG ( m, h ) the origin and all t -dimensionalsubspaces that contain it.Euclidean geometry has played a key role on multipleoccasions in the history of coding theory. The cyclic codes weexamine here have extensively been studied in many contextsfor several decades as well. For instance, it provided a classicexample of majority logic decodable codes in 1960’s [21]as well as remarkably high performance codes for moderniterative decoding [22], [23] in this century. In quantuminformation theory, Euclidean cyclic codes were studied asasymmetric quantum low-density parity-check codes [24], andquantum error-correcting codes that proved the potential of theentanglement-assisted formalism during the last decade [25],[26]. Our result on their minimum distances can be seen asa theoretical contribution to the study of this famous class ofcyclic codes.We also prove that these Euclidean cyclic codes have thebest possible tolerance against misalignment when used as theingredients of quantum synchronizable codes. Unfortunately,as we will show later when giving their minimum distances,the corresponding quantum synchronizable codes can not out-perform those based on primitive, narrow-sense BCH codes.Nonetheless, because of the mathematical similarity to thecase of projective geometry as well as the importance ofdetermining the minimum distance of a well-known cyclic UJIWARA AND VANDENDRIESSCHE: QUANTUM SYNCHRONIZABLE CODES FROM FINITE GEOMETRIES 7 code in general, we give all mathematical details in theremainder of this section.Let B be the set of t -dimensional subspaces of EG ( m, h ) .The incidence vector χ π of a t -dimensional subspace π ∈ B is defined in the same way as in PG ( m, h ) , so that the entryof each coordinate is if π contains the corresponding pointand otherwise. Define E m,t, h = h χ π | π ∈ Bi ⊥ to bethe dual of the vector space spanned by the incidence vectorsof t -dimensional subspaces in EG ( m, h ) . As in the case ofprojective geometry over finite fields, the cyclic group of order hm − acts regularly on the points in the case of EG ( m, h ) .Hence, by indexing the coordinates of incidence vectors by g , g , . . . , g hm − for a generator g of the cyclic group in thenatural order, E m,t, h can be seen as a cyclic code of length hm − . Proposition 3.16: E m,t, h is cyclic as a linear code.The cyclic code E m,t, h is one of the oldest efficientlydecodable codes. Its basic properties in this context can befound in a modern textbook [19, Section 13.8]. The followingis the explicit description of the generator polynomial of E m,t, h . Theorem 3.17 ([21]):
Let α be a primitive element in F hm .The generator polynomial g ( x ) of E m,t, h is g ( x ) = Y j ∈ I ′ m,t,h ( x − α j ) , where I ′ m,t,h = (cid:26) a ∈ N (cid:12)(cid:12)(cid:12)(cid:12) a ≤ hm − , max ≤ i ≤ h w h ( a i ) ≤ ( m − t )(2 h − (cid:27) . As in the case of C m,t, h from PG ( m, h ) , we show thatthe linear code E m,t, h is not only cyclic but also dual-containing and suitably nested while having good parametersand providing maximum synchronization recovery capabilitiesthrough Theorem 2.1 if t is in an appropriate range withrespect to m .The dual-containing property is a natural consequence of afundamental property of affine geometry. While this fact haslong been known among the finite geometry community, wegive a short proof for completeness. Lemma 3.18: If m +12 ≤ t ≤ m − , then E ⊥ m,t, h ⊆ E m,t, h . Proof:
It suffices to prove that for any pair π , π ∈ B of t -dimensional subspaces of EG ( m, h ) , the correspondingincidence vectors χ π , χ π ∈ E ⊥ m,t, h are orthogonal to eachother. Note that χ π and χ π are orthogonal to each other ifand only if the cardinality | π ∩ π | is even. Because t ≥ m +12 ,the intersection between χ π and χ π is either empty or anonempty subspace of AG ( m, h ) . Thus, | π ∩ π | is either or a positive integer power of as desired.The nested property of E m,t, h can be shown directlythrough their generator polynomials. Lemma 3.19: If ≤ t ≤ m − , then E m,t − , h ⊂ E m,t, h . Proof:
Let a ′ be the smallest integer such that max ≤ i ≤ h w h ( a ′ i ) = ( m − t )(2 h −
1) + 1 . Then, because ≤ t ≤ m − , we have a ′ = 2 h ( m − t ) + m − t − X i =0 (2 h − hi = 2 h ( m − t ) + 2 h ( m − t − − < hm − . Hence, by Theorem 3.17, the degree of the generator poly-nomial g t − ( x ) of E m,t − , h is strictly larger than that ofthe generator polynomial g t ( x ) of E m,t, h while g t ( x ) divides g t − ( x ) . Thus, E m,t − , h is strictly contained in E m,t, h .Proposition 3.16 and Lemmas 3.18 and 3.19 ensurethat E m,t, h possesses the properties of being cyclic, dual-containing, and nested, which are the minimum requirementsin Theorem 2.1. The remainder of this section investigates theparameters of E m,t, h as a code for standard error correctionand as a scheme for block synchronization recovery.Trivially, the length of E m,t, h as a linear code is exactlythe number of points, which is hm − . The dimension dim E m,t, h can be directly obtained through the followingformula that relates the dimension of Euclidean geometry tothat of projective geometry. Theorem 3.20 ([27]):
For positive integers m , h , t such that t ≤ m − , the dimension dim E ⊥ m,t, h of the vector spacespanned by the incidence vectors of t -dimensional subspacesin EG ( m, h ) is dim E ⊥ m,t, h = dim P m,t, h − dim P m − ,t, h − . Since the above theorem gives the dimension of thedual, dim E m,t, h is obtained simply by taking hm − − dim E ⊥ m,t, h . Lemma 3.21:
For positive integers m , h , t such that t ≤ m − , the dimension of E m,t, h is dim E m,t, h = 2 hm − dim P m,t, h + dim P m − ,t, h . Note that the exact values of dim P m,t, h and dim P m − ,t, h can be obtained by Theorem 3.8, allowing for computing dim E m,t, h for given m , t , and h .To prove the exact minimum distance of E m,t, h , we use thewell-know BCH bound on the minimum distance of a cycliccode. Theorem 3.22 (BCH bound for binary codes):
Let g ( x ) bethe generator polynomial of a cyclic code of length n andminimum distance d . Let n ′ be the smallest integer such that n divides n ′ − and α a primitive n th root of unity in F n ′ . Ifthere exist a nonnegative integer b and positive integer δ ≥ such that g ( α b + i ) = 0 for ≤ i ≤ δ − in F n ′ , then d ≥ δ .The proof of the BCH bound can be found in a standardtextbook in coding theory such as [11]. We show that theBCH bound is sharp for E m,t, h by explicitly constructing acodeword whose weight meets the lower bound.A hyperoval in a -dimensional subspace of AG ( m, h ) isa set of h + 2 points no three of which are contained in thesame -dimensional subspace. Such a configuration exists forall m and h . We show that a combination of a hyperoval and ( t − -dimensional subspace leads to a nonzero codeword ofminimum weight in E m,t, h . IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. X, NO. XX, MONTH YEAR
Theorem 3.23: E m,t, h is a linear [2 hm − , hm − dim P m,t, h + dim P m − ,t, h , (2 h − + 1)2 h ( m − t − − code. Proof:
It suffices to prove that the minimum distance d of E m,t, h is (2 h − + 1)2 h ( m − t − − . It is straightforwardto see that every positive integer a smaller than h ( m − t ) +2 h ( m − t − − satisfies the condition that max ≤ i ≤ h w h ( a i ) ≤ ( m − t )(2 h − . By Theorem 3.17, for all positive integer i ≤ h ( m − t ) +2 h ( m − t − − , the generator polynomial g ( x ) of E m,t, h has ( x − α i ) as its factors. Hence, by Theorem 3.22, we have d ≥ (2 h − + 1)2 h ( m − t − − . We construct a codewordof weight (2 h − + 1)2 h ( m − t − − . Take a hyperoval H in a -dimensional subspace of AG ( m, h ) . Without loss ofgenerality, we assume that H contains the origin. Let L bethe line at infinity and take a ( t − -dimensional subspace π in the hyperplane at infinity that is disjoint from L . Theunion of h + 2 parallel spaces {h p, π i | p ∈ H } is aset of (2 h + 2)2 h ( m − t − points whose incidence vector liesin the dual of the vector spaces spanned by the incidencevectors of t -dimensional subspaces of AG ( m, h ) . Removingthe coordinate corresponding to the origin gives a codewordof weight (2 h + 2)2 h ( m − t − − in E m,t, h .The above theorem may be seen as a precise result onthe parameters of the classic examples of majority decodableerror-correcting codes and finite geometric low-density parity-check codes as well. As in the case of C m,t, h based on thecomplement of projective geometry, the exact values of allparameters for given m , t , and h can be obtained by applyingTheorem 3.8 to this theorem. Table II lists the parameters of E m,t, h for some m , t , and h .It should be noted that, unlike the projective case, the lengthof this type of cyclic code is always a Mersenne number.When compared to the primitive, narrow-sense BCH codeof the same length, the generator polynomial of E m,t, h isof higher degree, which implies that E m,t, h has a smallerdimension. Now the BCH bound is a lower bound on theminimum distance of the BCH code. However, the abovetheorem shows that the minimum distance of E m,t, h actuallymatches the BCH bound. Hence, the cyclic codes based onEuclidean geometry are generally poorer in terms of infor-mation rate and minimum distance. Considering the decenterror-correcting performance reported in the literature, thepoorer minimum distance property seems to suggests that theerror-correcting codes based on Euclidean geometry benefitmore from decoding algorithms that are less sensitive to trueminimum distances such as the sum-product algorithm andmajority logic algorithm.Now, in the context of quantum synchronizable coding, wewould like as high misalignment tolerance as possible. As isthe case with projective geometry, we prove that any pairof cyclic codes taken from the nested chain E m, ⌈ m +12 ⌉ , h ⊂E m, ⌈ m +12 ⌉ +1 , h ⊂ · · · ⊂ E m,m − , h attains the trivial upperbound in this regard. Lemma 3.24:
Let g ( x ) and h ( x ) be the generator polynomi-als of E m,t, h and E m,t − i, h for a positive integer i ≤ t − re- TABLE IIS
AMPLE PARAMETERS OF E m,t, h FROM EG ( m, h ) . m h a t Length Dimension Minimum distance5 2 3 1023 748 235 2 4 1023 988 55 3 3 32767 28042 795 3 4 32767 32552 96 2 4 4095 3572 236 2 5 4095 4047 56 3 4 262143 249816 796 3 5 262143 261801 96 4 4 16777215 16490000 2876 4 5 16777215 16774815 177 2 4 16383 11728 957 2 5 16383 15473 237 2 6 16383 16320 57 3 4 2097151 1763104 6397 3 5 2097151 2068983 797 3 6 2097151 2096640 98 2 5 65535 55627 958 2 6 65535 64055 238 2 7 65535 65455 58 3 5 16777215 15742657 6398 3 6 16777215 16719003 798 3 7 16777215 16776487 99 2 5 262143 184848 3839 2 6 262143 242724 959 2 7 262143 259860 239 2 8 262143 262044 5 a When h = 1 , the parameters of E m,t, and C m − ,t, coincide. spectively. Define f ( x ) to be the quotient of h ( x ) = f ( x ) g ( x ) divided by g ( x ) . Then ord( f ( x )) = 2 hm − . Proof:
By Theorem 3.17, we have f ( x ) = Y j ∈ I ′ m,t − i,h \ I ′ m,t,h ( x − α j ) . Let j = 2 h ( m − t ) − and j = 2 h ( m − t ) − . It is easyto see that these two relatively prime integers are in the set I ′ m,t − i,h \ I ′ m,t,h . Write the minimal polynomials of α j and α j as M α j ( x ) and M α j ( x ) respectively. By Propositions3.13 and 3.14, we have ord( f ( x )) ≥ lcm (ord ( M α j ( x )) , ord ( M α j ( x )))= lcm (cid:18) hm − j , hm − , hm − j , hm − (cid:19) = 2 hm − . Since the order of a factor of the generator polynomial ofa cyclic code is at most the length of the code, we have ord( f ( x )) = 2 hm − as desired.The following theorem summarizes the results presented inthis subsection. Theorem 3.25:
Let m , h , and t be positive integers suchthat m +12 ≤ t ≤ m − . For every pair a l , a r of nonneg-ative integers such that a l + a r < hm − there exists aquantum synchronizable ( a l , a r ) - (cid:2)(cid:2) hm − a l + a r , hm − UJIWARA AND VANDENDRIESSCHE: QUANTUM SYNCHRONIZABLE CODES FROM FINITE GEOMETRIES 9 P m,t, h + 2 dim P m − ,t, h + 1 (cid:3)(cid:3) code that corrects atleast up to (2 h − + 1)2 h ( m − t − − phase errors and at leastup to (2 h − + 1)2 h ( m − t − − bit errors. Proof:
Apply Theorem 2.1 to E m,t, h and E m,t +1 , h withProposition 3.16, Lemmas 3.18, 3.19, and 3.24, and Theorem3.23. A routine calculation proves the assertion.IV. C ONCLUDING REMARKS
We constructed a family of quantum synchronizable codesthat correct both standard quantum errors and block synchro-nization errors. One type of our code exploits the complementstructure of projective geometry while the other type takesdirect advantage of Euclidean geometry without the origin.Both types of codes are proved to achieve the highest possibletolerance against misalignment. The results presented in thispaper enrich the variety of available quantum synchronizableerror-correcting codes.While cyclic codes are useful in both classical and quantuminformation theory, it is not easy to construct ones with largeminimum distances. A particularly difficult task is to preciselydetermine the parameters instead of bounding them from aboveor below. For instance, given a generator polynomial, it is avery challenging algebraic problem to give the exact minimumdistance of the corresponding cyclic code. In fact, preciseresults on cyclic codes with fairly large minimum distancessuch as Theorems 3.11 and 3.23 given in this paper are notcommon in the literature. Significant mathematical advancesin this aspect are much desired.It is also notable that the proof of Theorem 3.11 gives acomplete picture of the minimum weight nonzero codewordsof our cyclic codes C m,t, h based on projective geometry. Infact, its minimum weight nonzero codewords all come fromthe incidence vectors of ( m − t ) -spaces in PG ( m, h ) .In the case of E m,t, h , however, it appears more difficult toobtain a similar classification of the minimum weight nonzerocodewords. Although we did not identify all codewords ofweight (2 hm +2)2 m − t − − , we conjecture that every nonzerocodeword of minimum weight in E m,t, h is obtained in the wayshown in the proof of Theorem 3.23.Another notable point regarding our constructions for quan-tum synchronizable codes is that while Theorems 3.1 and 3.25were proved by using a pair of cyclic codes lying next to eachother in a chain of nested codes, we can employ any pair,such as C m,t, h and C m,t +2 , h , in the same chain to obtainanalogous theorems. The resulting quantum synchronizablecodes will have the same highest possible tolerance againstmisalignment, the same length, the same dimension, and thesame phase error correction capabilities as in Theorems 3.1and 3.25. The advantage is that these alternative codes requirefewer quantum interactions for detecting bit errors becausefewer stabilizer operators are involved. They have a drawbackof reduced bit error correction capabilities because the cycliccodes responsible for bit error detection will have smallerminimum distances.Note that the frameworks of quantum synchronizable codesgiven in [5], [6] implicitly assume that phase errors are morelikely, which is a reasonable assumption because asymmetry in error probability between bit errors and phase errors isexpected in actual quantum devices [28]. Our main results alsoput emphasis more on the minimum distance responsible forphase error correction than that for bit error correction. How-ever, in a situation where the quantum channel is very highlyasymmetric and introduces phase errors far more frequently,using even more asymmetric quantum synchronizable error-correcting codes may make more sense than employing thefairly asymmetric ones given in Theorems 3.1 and 3.25. Wehope that our results presented in this paper help advance thefield in various directions from both mathematical and physicalviewpoints. A CKNOWLEDGMENT
The authors thank the anonymous reviewers and AssociateEditor Alexei Ashikhmin for careful reading of the manuscriptand constructive suggestions.R
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Yuichiro Fujiwara (M’10) received the B.S. and M.S. degrees in mathematicsfrom Keio University, Japan, and the Ph.D. degree in information science fromNagoya University, Japan.He was a JSPS postdoctoral research fellow with the Graduate Schoolof System and Information Engineering, Tsukuba University, Japan, and avisiting scholar with the Department of Mathematical Sciences, MichiganTechnological University. He is currently with the Division of Physics,Mathematics and Astronomy, California Institute of Technology, Pasadena,where he works as a JSPS postdoctoral research fellow.Dr. Fujiwara’s research interests include combinatorics and its interactionwith computer science, quantum information science, and electrical engineer-ing, with particular emphasis on combinatorial design theory, algebraic codingtheory, and quantum information theory.