Quantum Cyclic Code of length dividing
In this paper, we study cyclic stabiliser codes over of length dividing for some positive integer . We call these -Frobenius codes or just Frobenius codes for short. We give methods to construct them and show that they have efficient decoding algorithms.
An important subclass of stabiliser codes are the linear stabiliser codes. For linear Frobenius codes we have stronger results: We completely characterise all linear Frobenius codes. As a consequence, we show that for every integer that divides for an odd , there are no linear cyclic codes of length . On the other hand for even , we give an explicit method to construct all of them. This gives us a many explicit example of Frobenius codes which include the well studied Laflamme code.
We show that the classical notion of BCH distance can be generalised to all the Frobenius codes that we construct, including the non-linear ones, and show that the algorithm of Berlekamp can be generalised to correct quantum errors within the BCH limit. This gives, for the first time, a family of codes that are neither CSS nor linear for which efficient decoding algorithm exits.
Successful implementation of quantum computing requires handling errors that occur while processing, storing and communicating quantum information. Good quantum error correcting codes are therefore a key technology in the eventual building of quantum computing devices, besides, perhaps more importantly, their theory provide some elegant mathematics. An important class of codes are the stabiliser codes , which not only captured the isolated examples constructed earlier [13, 14, 4, 11], but built a solid foundation for subsequent works [6, 3, 2].
Constructing stabiliser codes require handling the slightly non-standard symplectic inner product. The CSS construction [7, 15] gives one elegant and natural way, albeit with some loss of generality, to handle this difficulty. For this one needs a self-dual code classical code, or more generally two classical codes one contained in the dual of the other, thereby reusing the intuition built for classical codes. Another approach to the problem, again with some loss of generality, is to look at linear stabiliser codes . Linear stabiliser codes can also be characterised as linear classical codes over a quadratic extension of the base field [6, Theorem 3] [9, Lemma 18] which are Hermitian self-dual.
In this article, we study mainly cyclic stabiliser codes. Cyclic codes, being well studied classically, have recently been studied in detail [6, 16, 1, 9], mostly from the perspective of either self dual codes or Hermitian self dual codes. We explore another approach to simplify the symplectic condition, namely, we restrict our attention of cyclic codes of length dividing over .
In this article, we focus on cyclic stabiliser codes over the field whose lengths divide , for some positive integer . We call such codes -Frobenius codes, or just Frobenius codes, because of the key role played by the Frobenius automorphism. Restricting to such lengths, while constraining, is not that bad, as there is a healthy, i.e. almost linear, density of such lengths. In bargain, we get a simpler formulation of the isotropy condition, which helps in the analysis of these codes considerably. Furthermore, this simplicity of the isotropic condition allows us to extend the notion of BCH distance for these codes and give efficient decoding algorithms. Since none of the codes that we construct are CSS — all our codes are uniquely cyclic (See Section III for a definition) and by Proposition III.5 are not CSS — and some of them are non-linear, this gives a family of codes for which efficient decoding algorithms were not known before.
We study the subfamily of linear Frobenius codes in detail and completely characterise them (Theorems IV.4 and IV.6). This has two consequence, one negative and another positive. Firstly, over , we show that there are no -Frobenius linear codes when is odd (Corollary IV.5). This is a somewhat serious limitation of linear cyclic codes as the density of such lengths seems to be almost linear. Moreover, this impossibility is purely Galois theoretic unlike other known restriction that arise from sphere packing bounds or linear programming bounds.
On the positive side, the characterisation of linear Frobenius codes gives us ways to explicitly construct examples of linear Frobenius codes of lengths . Again, since the density of such lengths are also healthy, this technique give sizable number of explicit examples including the well studied Laflamme code. Table I give such examples for and lengths less than 100.
We give a brief overview of the notation used in this paper. For a prime power , denotes the unique finite field of cardinality . The product is a vector space over the finite field and an element in it is thought of as a column vectors. Fix a -dimensional Hilbert space . An orthonormal basis for is of cardinality . Fix one such basis and denote it by . As is standard in quantum computing, for an element in , denotes the tensor product . The set forms a basis for the -fold tensor product . A quantum code over of length is a subspace of the tensor product . There is by now a significant literature on quantum codes [10, 8, 6].
Let denote the primitive -th root of unity . For and in , define the operators and on as and respectively. The operator can be thought of as a position error and as a phase error. In a quantum channel, both position errors and phase errors can occur simultaneously. These are captured by the Weyl operators .
For elements and of the vector space the joint weight is the number of positions such that either or is not zero. The weight of the Weyl operator is the joint weight . Occurrence of a quantum error at some positions is modelled as the channel applying an unknown Weyl operator of weight on the transmitted message.
An important class of quantum codes are the class of stabiliser codes . One can study stabiliser codes by studying the isotropic sets under the symplectic inner product. For any two vectors and of , define the symplectic inner product as the scalar of . A subset of is called totally isotropic , or just isotropic, if for any two elements and of , .
Isotropic subspaces of are closely related to stabiliser codes. Calderbank et al [5, 6] were the first to study this relation when the underlying field is . Later, this was generalised to arbitrary fields [3, 2]. We summaries these results in a form convenient for our purposes.
Let be a isotropic subspace of for some positive integer . Let be either the primitive -th root of unity or , depending on whether is odd or even respectively. Then, the subset of unitary operators forms an Abelian group. Furthermore, the set of vectors invariant under the operators in forms a quantum stabiliser code and the operator is the projection to it.
Let be a subspace of . By the centraliser of , denoted by , we mean the subspace of all in , such that , for all in . We have the following theorem on the error correcting properties of the stabiliser codes.
Let be a isotropic subspace of and let be the associated stabiliser code. Then the dimension of the subspace is at most . If has dimension for some then the centraliser , as a vector space over , is of dimension and the code , as a Hilbert space, is of dimension . Furthermore, if the minimum weight is then can detect up to errors and correct up to errors.
Let be a stabiliser code associated with an dimensional totally isotropic subspace of . By the stabiliser dimension of we mean the integer . Similarly, we call the weight the distance of . In this context, recall that the stabiliser code associated to the isotropic set is called -pure, if the minimum of the joint weights of non-zero elements of the centraliser is . It follows from Theorem II.2 that a -pure code is of distance at least . A stabiliser code over of length , stabiliser dimension and distance is called an code.
Iii Quantum Cyclic codes
In this section we define quantum cyclic codes and study some of its properties. Fix a prime and a positive integer coprime to for the rest of the section. Let denote the right shift operator over , i.e. the operator that maps to . Consider the unitary operator defined as . Recall that a classical code over is cyclic if for all code words , its right shift is also a code word. Motivated by this definition, we have the following definition for quantum cyclic codes.
A quantum code is cyclic if for any vector in , the vector is in .
Let be a subspace of . We say that is simultaneously cyclic if for all in , is also in . Stabiliser codes with simultaneously cyclic isotropic sets were first studied by Calderbank et al [6, Section 5] and was taken as the definition of cyclic codes in subsequent works [16, 1, 9]. In this context, we show that for stabiliser codes, simultaneous cyclicity and our definition of cyclicity coincide.
An isotropic subset of is simultaneously cyclic if and only if the associated stabiliser code is cyclic.
For a code with projection operator , it is easy to verify that is cyclic if and only if . Let be an isotropic subset of and let be the associated stabiliser code.
Assume that is cyclic. From Theorem II.1, the projection operator to is given by , where . Notice that . Therefore, the set should be simultaneously cyclic, otherwise the support of will not match with that of .
Conversely, if is simultaneously cyclic, then we have is in for all in . The inverse of the shift operation is just . Therefore, and hence the scalars are also preserved. Thus, and as a result, is cyclic. ∎
Let denote the cyclotomic ring of polynomials modulo . When dealing with cyclic codes, it is often convenient to think of vectors of as polynomials in by identifying the vector with the polynomials . We use the bold face Latin letter, for example , etc, to denote vectors and the corresponding plain face letter, , respectively, for the associated polynomial. Recall that, classical cyclic codes are ideals of this ring . In the ring , the polynomial has a multiplicative inverse namely . Often, we write to denote this inverse. Notice that for any two vectors and in , if and denote the corresponding polynomials in , then the coefficient of in the product is the inner product , where is the right shift operator. An immediate consequence is the following.
Let be a simultaneously cyclic subset of . Then is isotropic if and only if for any two elements and , the corresponding polynomials satisfy the condition
Let be a simultaneously cyclic subspace of . Define and to be the projections of onto the first and last coordinates respectively, i.e. and . Since is simultaneously cyclic, and are cyclic subspaces of and hence are ideals of the ring . Let be the factor of that generates . Since is an element of , there exists a polynomial in such that . If this is unique then we say that is uniquely cyclic and call the pair of polynomials, a generating pair for . We have the following proposition.
A simultaneously cyclic subspace of is uniquely cyclic if and only if for every element in , . If is uniquely cyclic generated by the pair , then every element of is of the form for some in .
For a CSS code, the underlying isotropic set is a product of two -length classical codes over In particular, elements and for and in and respectively belong to . Therefore, we have the following proposition as a consequences of Proposition III.4.
Any uniquely cyclic stabiliser code is not CSS unless it is of distance .
For uniquely cyclic codes the isotropy condition in Proposition III.3 can be simplified as follows.
Let be a simultaneously cyclic subspace of with generating pair . Then is isotropic if and only if modulo . Moreover, any pair belongs to if and only if modulo .
Consider a quadratic extension of obtained by adjoining a root of some quadratic irreducible polynomial over . Identify the product with the the vector space by mapping a pair of vectors to the vector . Similarly for the cyclotomic ring , identify the product ring with the cyclotomic ring . Let be any isotropic subspace of . The associated stabiliser code is said to be linear  if under the above identification is a subspace of . Isotropic subspaces of associated to linear stabiliser codes are classical cyclic codes of length over . Thus the following proposition follows.
Let be an isotropic simultaneously cyclic subspace of the product . The associated stabiliser code is linear if and only if is an ideal of the cyclotomic ring . Furthermore, if is linear then the centraliser is also an ideal of .
It follows from the theory of classical codes that both and are ideals generated by factors of over . In this context, we make the following definition.
Definition III.8 (BCH distance).
Let be a factor of the polynomial over the field , coprime to . The BCH distance of the polynomial is the largest integer such that the consecutive distinct powers , are roots of , for some primitive -th root .
Recall that, the distance of a classical cyclic code is at least the BCH distance of its generating polynomial. In the setting of stabiliser codes, the distance is related to the minimum joint weight of elements of (Theorem II.2). Motivated by this analogy, we define the BCH distance of linear stabiliser codes as follows.
Let be a isotropic subset of associated to a linear cyclic stabiliser code . The BCH distance of is the BCH distance of the generator polynomial of the centraliser .
We have the following theorem which follows from Theorem II.2.
Let is be any linear cyclic stabiliser code of BCH distance . Then it is -pure and hence has distance at least .
Iv Linear cyclic codes of length dividing
In this section, we study linear cyclic stabiliser codes over whose length divides . The main motivation to restrict our attention to lengths of this form is captured in the following proposition.
If the integer divides , for some positive integer then in the cyclotomic ring is . Therefore, for every polynomial over any extension of we have is .
The above-mentioned property simplifies the isotropy condition for polynomials considerably and allows us to completely characterise all linear cyclic codes of such lengths.
Let be an extension of degree . When dealing with cyclic quantum codes of length , we use to denote the cyclotomic ring . The extension ring is then the cyclotomic ring . Linear codes are associated with quadratic extension and identification of the pair of vectors with maps its isotropic set to an ideal of .
Let be the isotropic ideal associated to a linear cyclic stabiliser code over of length dividing . Then is uniquely cyclic.
Let be the quadratic extension such that is an ideal of the cyclotomic ring . Let be the minimal polynomial of over .
Recall that the projection of onto the first co-ordinates forms a classical cyclic code over and hence is generated by a factor of . Suppose there exist two distinct elements and in . Let so that is also in . We prove that the polynomial modulo . By the Chinese remaindering theorem, it is sufficient to prove separately that all roots of and are roots of .
We now show that every root of is also a root of . Since belongs to , if the code is linear then must also belong to where is a root of the quadratic polynomial . Any element in is of the form , where and are polynomials in . Hence, and every root of is also a root of . ∎
Consider the Frobenius automorphism on a degree extension which maps any element in to . This can be naturally extended to polynomials over and therefore on as follows: For a polynomial where are in , is defined as . We call this the Frobenius involution.
Constructing linear cyclic codes correspond to constructing generators for the associated isotropic ideal. We make use of the following Galois theoretic lemma to characterise such generators.
Let the integer divide for some positive integer .
Any irreducible factor of over other than the factors or has even degree.
Let be any irreducible factor of over whose degree is divisible by for some positive integer . Over the extension field , splits into irreducible factors such that .
Consider any irreducible factor of over other than or . Let be the degree of . Then, the splitting field of over is . Consider any root of in . The Frobenius automorphism is a field automorphism of and . Notice that is an -th root of unity and divides . Hence and . Since is neither nor , we have and hence . As a result, the roots of comes in pairs; for every root its inverse is also a root. Hence, the degree of should be an even number. ∎
Consider any irreducible factor of of degree for some positive integer . Its splitting field therefore, contains . Any irreducible factor of over should be of degree equal to the degree of the extension which is . The Frobenius being a field automorphism of , should map these factors to each other. Further, order of in is . Thus over . ∎
Consider the extension field and let be any ideal of . The following theorem gives a necessary condition for it to be isotropic and hence give a linear cyclic code.
Let be a quadratic extension of . Let divide and be an isotropic ideal of . Then is even and the ideal is generated by the product polynomial where and are two coprime factors of satisfying the following condition.
is any factor of over which contains both and as factors.
is any factor of over , such that for any irreducible factor of over , divides if and only if does not.
From Lemma IV.2 it follows that is uniquely cyclic. Let be a generating pair for where and are polynomials over . Then the polynomial is an element of the ideal . It follows from the linearity of that the polynomial is also in . However, the set is uniquely cyclic. Using Proposition III.4, there is a polynomial in such that
Let be the minimal polynomial of over where and are elements of . Comparing the coefficients of in Equation 1, we have
When divides , for any polynomial in the cyclotomic ring , is just . Since is isotropic, from Equation 2 and Proposition III.3 its follows that . The polynomial is invertible modulo . As a result we have,
Let be any irreducible factor of over and be the extension field . From Equation 3 we have, is a root of the polynomial over the extension field . If possible, let be an odd integer . Since divides , . Using Equation 4 we get modulo and hence is an element of the sub field of . However, this is a contradiction, since the polynomial is irreducible over . Therefore, must be even.
Recall that is a root of the polynomial over the extension field . This implies that the extension field contains . Therefore, degree of must be even and must have as factors all the odd degree irreducible factors of over . By Lemma IV.3, these odd degree factors are just and . Thus satisfy property 1 of the theorem.
Consider the polynomial . Clearly is coprime to . We claim that generates the ideal . To see this, notice that as a subspace of is uniquely cyclic and is generated by the pair , where modulo (using Equation 2). Therefore, as an ideal is also generated by the polynomial which is the product . We claim that polynomial thus constructed satisfies the properties mentioned in the theorem. Any irreducible factor of over the field is of even degree and hence factorises as over . Recall that is a root of . As a result, is either or . Now, divides if and only if modulo is zero. Therefore, divides if and only if . The polynomial has coefficients in and hence . As a result by the third property of Proposition .1, if and only if . For each pair and , exactly one of them divide depending one whether is or modulo . This proves the theorem. ∎
A corollary of the above theorem is the following impossibility result.
Let be any integer that divides , where is odd. Then there does not exist any linear cyclic stabiliser codes of length over .
For example, are the numbers less then hundred that divide for some odd . Hence there is no binary linear cyclic code of such lengths.
The next theorem shows that the conditions in Theorem IV.4 are also sufficient to construct isotropic ideals of . This gives us a way of constructing linear cyclic stabiliser of length dividing . This theorem directly follows from a more generalised construction given in Theorem V.1 and Theorem V.5.
In the rest of the article, we refer to cyclic stabiliser codes whose length divide as -Frobenius codes. For linear -Frobenius codes, we call the factorisation characterised above as the canonical factorisation associated to the code.
Let be a linear -Frobenius code over with canonical factorisation . The stabiliser dimension of the code is . The centraliser of is the ideal generated by and hence the BCH distance of is BCH distance of .
V Generalisation to nonlinear codes
We have already shown that if divides for some odd integer then no linear code of length exists. In this section we show how to construct nonlinear codes of such length. The construction is a generalisation of Theorem IV.6. The major difference is that the extension of is no longer restricted to be quadratic.
Let divide and be a degree extension of . Let and be co-prime factors of satisfying the following properties.
is any factor of over which contains all the the irreducible factor of over whose degree is not divisible by .
is any factor of over such that for any irreducible factor of over , divides if and only if none of the factors divide i.e. .
Fix any nonzero in and let be the polynomial, uniquely defined by Chinese remaindering, as follows.
Then is a polynomial in and the uniquely cyclic subspace generated by is isotropic.
The proof of this theorem, involves verifying certain equations modulo . Almost always we do this in by verifying the said equation separately modulo and each irreducible factor of . Then by Chinese remaindering, we have the said equation modulo . We call this the Chinese remainder verification.
Let us call the polynomial as . From the definition of the factor it follows that over , the polynomial splits as .
The polynomial is a polynomial in
It is sufficient to prove that . Notice that since , . Using Proposition .1, it is sufficient to show that . Since and for all , applying on both side we get for all . By Chinese remainder verification the claim follows. ∎
From Proposition IV.1 we have, . We know, . Since , applying on we get . Hence by Chinese remaindering, . ∎
The uniquely cyclic subspace generated by the pair is isotropic.
Let be an irreducible polynomial over and be roots of . Fix , to be the extension and in Theorem V.1 and let be the corresponding isotropic subspace. Then the image of under the map is an ideal of the cyclotomic ring and its generator is given by the polynomial where satisfies the properties in Theorem IV.4. Moreover the centraliser also maps to an the ideal generated by .
We know from Theorem V.1 that the polynomial defined by the following
belongs to the ring and the uniquely cyclic subspace generated by the pair is isotropic.
To prove that maps to an ideal it is sufficient to show that for any element in there exists another element in such that . We claim that we can always choose to satisfy this condition.
The cyclotomic polynomial is product of , and . Using equation 6 and the fact that , it is straight-forward to verify the claim separately modulo , and . Then by Chinese remaindering conclude that it is true modulo . ∎
Equation 6 implies that for any irreducible factor of over
Now is a generator of as an ideal and equation 7 imply that if then is zero and if then is which is nonzero. Therefore and is also a generator of .
Let be the ideal generated by . Since , the ideal is also generated by . We know that (see proof of Claim V.3). By Proposition III.6 it can be verified that any element of the form belongs to the centraliser. Since itself is an ideal, is a subideal of . To show that is actually we show that they have same cardinality. The cardinality of is which is equal to , since . On the other hand cardinality of is . Hence by Theorem II.2 cardinality of is . ∎
As before, we call as the canonical factorisation associated with the above mentioned -Frobenius codes. We also call the BCH distance of to be the BCH distance of .
Let be the canonical factorisation associated with a -Frobenius code as in Theorem V.1. The stabiliser dimension of is . If the BCH distance of is then is -pure and hence has distance at least .
Let be the polynomial corresponding to such that the isotropic subspace of is generated by the pair as in Theorem V.1. Since any element in is of the form , the number of distinct values can take is the cardinality of the ring which is . Hence by Theorem II.2 we conclude that the stabiliser dimension of is
To prove the lower bound on distance we first need the following result
Any element in the centraliser is of the form for some polynomials and over such that and are coprime.
Let be the set of all pairs where and are polynomials over such that is coprime to . It follows from Proposition III.6 that the set is contained in . However, the cardinality of is the product of the cardinalities of the rings and which is . By Theorem II.2, cardinality of itself is . Hence is equal to the set . ∎
Notice that the joint weight of a pair is equal to the weight of as a polynomial over . We know, for any factor of , weight of any polynomial over which is a multiple of , is at least the BCH distance of . Therefore, to prove that is -pure it is sufficient to show that for any element in the centraliser the polynomial is a factor of . By Claim V.8 we know that there exists a polynomials such that . Since divide and , it follows that . ∎
As a demonstration of our construction we list (Table I) some explicit examples of codes where the characteristic