Quantum Bicyclic Hyperbolic Codes

Bicyclic codes are a generalization of the one dimensional (1D) cyclic codes to two dimensions (2D). Similar to the 1D case, in some cases, 2D cyclic codes can also be constructed to guarantee a specified minimum distance. Many aspects of these codes are yet unexplored. Motivated by the problem of constructing quantum codes, in this paper, we study some structural properties of certain bicyclic codes. We show that a primitive narrow-sense bicyclic hyperbolic code of length n^2 contains its dual if and only if its design distance is lower than n-Δ, where Δ=O(√(n)). We extend the sufficiency condition to the non-primitive case as well. We also show that over quadratic extension fields, a primitive bicyclic hyperbolic code of length n^2 contains Hermitian dual if and only if its design distance is lower than n-Δ_h, where Δ_h=O(√(n)). Our results are analogous to some structural results known for BCH and Reed-Solomon codes. They further our understanding of bicyclic codes. We also give an application of these results by showing that we can construct two classes of quantum bicyclic codes based on our results.



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1 Introduction

Cyclic codes are an important class of error-correcting codes. Many popular codes, such as BCH codes and Reed-Solomon codes, are cyclic codes. Cyclic codes with guarantees on the minimum distance of the code are easy to construct. Many subclasses of cyclic codes also have efficient decoders making them suitable for practical applications. For quantum error correction, a classical code can be used to construct quantum code [14, 5, 4, 2] if the code contains its (Euclidean or Hermitian) dual. Using these constructions many (cyclic) quantum codes have been proposed [8, 7, 10]. Grassl et al. gave a simple test for identifying cyclic codes that contain their duals [8]. Steane [15] gave a condition, based on the designed distance, to check whether a primitive binary BCH contains its Euclidean dual. Subsequently, Aly et al. [1] extended this result to the higher alphabet and non-primitive codes. They proved that a primitive BCH code of length contains its dual when its design distance is less than . These results are based on significant structural results of cyclic codes. However, most of the previous work has been limited to one dimension [16, 13, 17, 11, 6, 12]

, even though classically, cyclic codes have been generalized to higher dimensions. Two dimensional (2D) cyclic codes, also called bicyclic codes, are a generalization of one-dimensional cyclic codes to two dimensions. In the case of one-dimensional cyclic codes, codewords can be considered as vectors. The codewords of bicyclic codes can be viewed as matrices. Imai

[9] introduced a general theory of 2D cyclic codes. Since then, there has been extensive work on bicyclic codes, see [3] for a good overview of related work and references. However, there does not appear to be much work on quantum bicyclic codes. There are many significant differences between cyclic and bicyclic codes which makes the analysis of bicyclic codes much more challenging than cyclic codes. The characterization of 2D cyclic codes is somewhat more complex than the cyclic codes. For instance, bicyclic codes do not have a unique generator polynomial, unlike the cyclic codes. Furthermore, the division of polynomials by bivariate polynomials does not lead to unique remainders. All these reasons motivate our study of quantum bicyclic codes. In this paper, we focus on a class of bicyclic codes called hyperbolic codes, see [3]. (Note that the term hyperbolic codes also refers to a class of topological quantum codes.) Our main contributions are as follows:

  1. We give necessary and sufficient condition for a bicyclic hyperbolic code to contain its Euclidean dual.

  2. For a bicyclic hyperbolic code, defined over a quadratic extension field, we also give a necessary and sufficient condition for it to contain its Hermitian dual.

  3. We construct new quantum bicyclic codes.

Our analysis of the cyclotomic cosets of bicyclic codes could be of independent interest and use in the study of cyclic codes over higher dimensions. Our paper is organized as follows. After a brief review of the necessary background in Section 2, we prove structural results on bicyclic codes in Sections 3 and 4. We conclude with a brief discussion on the significance of these results and the scope for future work.

2 Background

Let be a prime power, and denote a finite field of elements. We denote by the set of integers .

2.1 Bicyclic codes

A linear code is a -dimensional subspace of . A bicyclic code of length can be viewed as a subspace of . We give a quick review of bicyclic codes; readers interested in more details should refer to [3, 9].

Definition 1 (Bicyclic codes).

Let and a linear code of length over a field , whose codewords can be written as two dimensional arrays of size . If is closed under both circular right shift of columns and circular down shift of rows, then is called a bicyclic code of length over .

Let be a codeword of a bicyclic code of size . Then we can write as follows:

We denote by the codeword obtained by circular right shift of columns of and the codeword obtained by circular down shift of rows of .

Polynomial representation

As in the case of 1D cyclic codes, it is convenient to represent codewords as polynomials over . Let denote the ring of polynomials in two variables over . Codewords of a bicyclic code can be represented as bivariate polynomials in . We will call the polynomial associated with a codeword as a code polynomial. The code polynomials of a bicyclic code of size will be of the following form.

Then the code polynomials corresponding to codewords and will be

Linearity of bicyclic codes implies that for any code polynomial in a bicyclic code , and we also have in . Therefore, bicyclic codes can be viewed as ideals in the quotient polynomial ring .


It is well know that when , a 1D cyclic code of length over is completely characterized by a unique monic generator polynomial where divides . Due to lack of direct generalization of polynomial division from 1D to 2D, We cannot always characterize bicyclic codes by a unique generator polynomial. Therefore, bicyclic codes are better analyzed in terms of the common zeros of the code polynomials. Given a bicyclic code of length over , the common zeros of all the codeword polynomials, and completely characterize the bicyclic code when is co-prime to both and . Any polynomial of length which vanishes at these common zeros is a codeword of that bicyclic code. Common zeros will be of the form where and are the and primitive roots of unity. The set of all such possible zeros is


It is easy to keep track of zeros just by the exponents of and . Therefore the defining set of a bicyclic code is given as follows:


Observe that since the codewords are over the field and and may lie in an extended field of , If is in the defining set of code then all the points of the form for , should also be in the defining set of . The set of all such points is called the -ary cyclotomic coset of modulo and and denoted as .


2.2 Bicyclic hyperbolic codes

Definition 2 (Bicyclic hyperbolic codes).

Let and for . A bicyclic code of length over is called a hyperbolic code with designed distance if the defining set of is of the following form


where , . is the designed set of which completely characterizes the code.

In the above definition, there is a freedom to choose the values of and . A bicyclic hyperbolic code with designed distance is guaranteed to have a minimum distance greater than or equal to . Our definition of bicyclic hyperbolic code is slightly different from the definition given in [3]. Analogous to one dimensional case we say the hyperbolic code is primitive if and narrow-sense if . In this paper we focus on the narrow-sense primitive bicyclic hyperbolic codes with . In the rest of this paper, We denote a bicyclic hyperbolic code of design distance by .

3 Euclidean Dual Containing Bicyclic Codes

The Euclidean inner product of two elements in is defined as follows:


The Euclidean dual code of a linear bicyclic code is defined as


If is bicyclic, then is also bicyclic and its zeros can be given in terms of the zeros of . When , the defining set of is completely characterized in terms of the defining set of [9]. Suppose is the defining set of and the defining set of . Then




A simple test to check if a cyclic code contains its dual in terms of its defining set was given in [8]. The same condition holds for bicyclic codes, which we are giving in the following lemma. Although straightforward, we give the proof for completeness.

Lemma 3.

Let be the bicyclic code of length over such that and let be the defining set of . Then contains its Euclidean dual if and only if


where .


Let and be two bicyclic cyclic codes with defining sets and . The code is contained in means that all the codewords that are in are also in . This is equivalent to saying that the defining set (or common zero set) of is contained in that of i.e., . From Eq. (8), is the defining set of dual code . Hence, if and only if . This is possible if and only if . ∎

The above condition for Euclidean dual containing can be further simplified for bicyclic hyperbolic codes with a designed set since the defining set is completely characterized .

Lemma 4.

Let be a bicyclic code of length over such that . Let and be the designed set and defining set of respectively. Then if and only if


where .


By Lemma 3, if and only if . To prove the lemma it suffices to show that if and only . Since if , then Eq. (12) holds. Suppose now that Eq. (12) holds. If , then . As is the union of cyclotomic cosets, it follows that . From Eq. (4), we have , it follows that . ∎

The following theorem gives an easy test, based on the designed distance, to check if a primitive narrow-sense hyperbolic code contains its Euclidean dual. In the proof of the following theorem 5 we will use lemma 6, which is stated after the theorem 5 to understand the motivation for lemma 6.

Theorem 5 (Euclidean dual containing bicyclic codes).

A primitive narrow-sense hyperbolic code of length over , where and , contains its Euclidean dual if and only if the design distance satisfies , where


Let . First, we show the sufficiency of for to be dual containing. For proving when , it is enough to show that contains . Since , this implies . The designed set for is given below:


By Lemma 4, we need to show Eq. (12) holds i.e., for . By the definition of and , for every the corresponding points must be in for all . Therefore if and only if the following inequality holds.


Observe that all the points satisfy . Therefore, the -ary expansions of and must satisfy the following constraint: If -ary coefficients of are equal to zero for and , then -ary coefficients of must be equal to zero for . Therefore all the points in must be of the following form, for some .


Based on Eq.(16), let us partition the points in into disjoint sets as follows.


Observe that, based on the definition of , for every point also belongs to . Therefore, it suffices to show the inequality in Eq. (15) holds for the points in where . This implies, from Eq. (16) and (17), instead of considering all points in , it is enough to consider just the following points:


Now let us try to lower bound the value of for , and .

  1. When is even: . From Lemma 6, we have the following equality.

  2. When is odd: . From Lemma 6, it follows that


This shows that every point satisfies and therefore cannot be in and thus satisfying Eq. (12). Next we show that is a necessary condition for to be dual containing. Seeking a contradiction let us assume that for some .

  1. When is even: consider the point . Since , from the definitions of (14) and ,


    Since , for we have the following equality.


    Therefore from (21) and (1), we can conclude that .

  2. When is odd: consider the point . Since ,


    If , then we have equationparentequation


    Hence, .

In both the above cases, we have . This implies, by Lemma 4, cannot contain giving us the desired contradiction. ∎

Now let see the structural results on the cosets in two dimensions which are used in the above theorem 5.

Lemma 6.

Suppose , and the sets and are as follows where .


Let , where . equationparentequation


Let . By equation (25), we have , and therefore should have the following -ary expansions.


where . Note that need not be nonzero here. Correspondingly will have the following -ary form.


The -ary expansions of and are obtained by taking the right circular shift of -ary expansions of and respectively.

  1. (30)

    When , -ary coefficient of is equal to . Hence If , then


    If the minimum value of is less than , then must be equal to , equivalently, must be equal to . Given that , implies and . Under these conditions minimum value of occurs when which is equal to .

  2. equationparentequation


    Let us assume that the minimum value of is less than . Let us check if our assumption is a valid one and if so, we will find which satisfy our assumption. If then is greater than or equal to . From Eq. (25), for some . Hence is greater than or equal to . Therefore, the product is greater than . When , is greater than or equal to and is greater than or equal to which implies the product is greater than or equal to . Combining this with , we get . we also have the inequality . This implies the possible values of are when is odd and when is even.

    1. When , equationparentequation


      Since is less than , at least on of the must be equal to zero. Otherwise will be greater than . This implies either or must be greater than or equal to and the product cannot be less than .

    2. When , equationparentequation


      Since is less than , at least on of the must be equal to zero. Therefore, from the above equations, minimum value of occurs when , and the minimum value is equal to

  3. equationparentequation


    Assume that the minimum value of is less than . Let us check if our assumption is a valid one. From equation (35), we can say that and . This, together with our assumption, implies must be less than . Additionally, the inequalities and imply and . Subsequently the following inequality holds true.


    Therefore our assumption is not a valid one and the minimum value of cannot be less than .

The cases i and iii imply Eq. (27a) and the case ii implies Eq. (27b). ∎

We can extend theorem 5 to the case of non-primitive narrow-sense bicyclic hyperbolic codes. The following corollary gives a sufficiency condition to verify if a narrow-sense bicyclic hyperbolic code contains its Euclidean dual code.

Corollary 7.

Suppose . Narrow-sense bicyclic hyperbolic code of length over contains its Euclidean dual if the design distance satisfies , where