Construction of optimal Hermitian self-dual codes from unitary matrices

11/24/2019 ∙ by Lin Sok, et al. ∙ 0

We provide an algorithm to construct unitary matrices over finite fields. We present various constructions of Hermitian self-dual code by means of unitary matrices, where some of them generalize the quadratic double circulant constructions. Many optimal Hermitian self-dual codes over large finite fields with new parameters are obtained. More precisely MDS or almost MDS Hermitian self-dual codes of lengths up to 18 are constructed over finite fields _q, where q=3^2,4^2,5^2,7^2,8^2,9^2,11^2,13^2,17^2,19^2. Comparisons with classical constructions are made.

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I Introduction

MDS codes form an optimal family of classical codes. They are closely related to combinatorial designs and finite geometry and have many applications in both theory and practice.

Self-dual codes are one of the most interesting classes of linear codes that find various applications in cryptographic protocols (secret sharing schemes) and combinatorics. They have close connections with group theory, lattice theory and design theory. It is well known that Euclidean binary self-dual codes are asymptotically good [27]. Constructions of Euclidean self-dual codes over large finite fields were given by many authors [1, 3, 11, 12, 13, 14, 15, 24, 25, 26, 32].

There have been a lot of works on Euclidean self-dual codes but less results have been known for the Hermitian case. The motivation of studying Hermitian self-dual (self-orthogonal) codes over was due to their connection to ary quantum stabilizer codes [2]. Quaternary Hermitian self-dual codes were considered by MacWilliams et al.[28] where they first gave the classification of length up to Following the work [28], Conway et al. [8] completed the classification of quaternary Hermitian self-dual codes of length up to Later Huffman [19, 20, 21]classified the extremal quaternary Hermitian self-dual codes of length up to . In 2011, Harada et al. [18] classified all quaternary Hermitian self-dual codes of length

There are two well-known constructions of Hermitian self-dual codes over large finite fields; the quadratic double circulant construction by Gaborit [11] and the building-up construction by Kim et al. [25].

Recently Tong and Wang [34] have constructed all MDS ary Hermitian self-dual codes of all lengths less than or equal to from the generalized Reed-Solomon codes.

In this paper we first study unitary groups over finite fields. We provide methods to construct unitary matrices and apply them to construct Hermitian self-dual codes over finite fields of prime power orders. We obtain many new optimal codes, more precisely MDS or almost MDS Hermitian self-dual codes of lengths up to 18 are constructed over finite fields with sizes . Further more with the same lengths, our method can also be applied efficiently for any being a square greater than . Some MDS and almost MDS as well as optimal Hermitian self-dual codes with new parameters are summarized in Table II and Table III.

For rather small values , we construct optimal Hermitian self-dual codes up to length (up to length for ). Numerical results show that our constructions perform better than the quadratic double circulant contruction [11] and the building-up construction [25], for example a Hermitian self-dual code over obtained from our construction has parameters which are better than those of [11] and Hermitian self-dual codes with parameters over respectively are better than those of [25] with the parameters over being new in [34]. Furthermore, over , we obtain MDS Hermitian self-dual codes with parameters which are better than [17] while the parameters over are new in [34]. On the one hand, our construction has no restriction on lengths like in [11] and thus more parameters are available. On the other hand, our method is easily applicable to construct codes of large lengths without going any recursive step like in [25], which makes the code construction faster. All the computations are done with Magma [5].

The paper is organized as follows: Section II gives preliminaries and background on self-dual codes as well as studies the unitary group over finite fields. Section III gives different constructions of Hermitian self-dual codes. Section IV studies matrix product codes which are Hermitian self-dual. Section V provides a method to embed a self-orthogonal code into a self-dual code. Section VI describes parameters of different constructions and makes comparisons among them. We end up with some concluding remarks in Section VII.

Ii Preliminaries

Ii-a Self-dual codes

A linear code of length over is a -dimensional subspace of . An element in is called a codeword. The (Hamming) weight wt

of a vector

is the number of non-zero coordinates in it. The minimum distance (or minimum weight) of is . For and in , their Euclidean and Hermitian inner product are defined respectively by

We say that is Hermitian orthogonal to if For we denote

For we denote The Euclidean (resp. Hermitian) dual of , denoted by (resp. ) is the set of vectors orthogonal to every codeword of under the Euclidean (resp. Hermitian) inner product. A linear code is called Euclidean (resp. Hermitian) self-orthogonal if (resp. ). A code is called Euclidean (resp. Hermitian) self-dual if (resp. ).

It is easy to verify that It is well known that a self-dual code can only exist for even length. If is an code, then from the Singleton bound, its minimum distance is bounded by

A code meeting the above bound is called Maximum Distance Separable (MDS) code and is called almost MDS if its minimum distance is one unit less than the MDS case. A code is called optimal if it has the highest possible minimum distance for its length and dimension and thus an MDS code is optimal.

Ii-B Unitary group over finite fields

As we will see in the next sections, a unitary group over finite fields plays a key role in our construction of Hermitian self-dual codes.

The unitary group of index over a finite field elements with is defined by

where is the matrix obtained from by taking the conjugate of all entries of A, that is, if then

The order of the group was determined by Wall [35] and is given as follows

(1)

In what follows, we present some elements used to generate a unitary group. Let for some prime and some positive integer Let if and otherwise. Let such that

(2)

and , if , where is the canonical basis of .

Define two linear maps

(3)

Denote

where is the permutation group of elements.

Lemma 1

Let There exist solutions for the system of equations defined by

(4)

Proof: Since we can choose and it is enough to prove that there exist such that

(5)

There exist such that the first equation of the system (5) holds. Now we have that and thus the second equation of the system also holds.  

Remark 1

It should be noted that are solutions to the system (2). However with these values, the linear map is just the identity map and thus it is out of interest.

Proposition 1

The group is a subgroup of

Proof: Obviously for any , we have and hence is Hermitian orthogonal. We also have since it is a matrix with all entries in and thus is Hermitian orthogonal . Finally under the conditions given in Eq. (2) we have that satisfies , where is the Kronecker symbol, and it is thus Hermitian orthogonal.  

The orders of the subgroup are computed using Magma [5] and given in Table I as well as compared with the orders of the unitary group given in Eq. (1).

TABLE I: The orders of computed in Magma [5] compared with those of Eq. (1) for .

We have already seen that the order of unitary groups in Table I grows very fast when the dimension becomes larger. The memory space for storing such matrices will be a challenging problem and visiting all the elements in the group is not possible say for . To search for optimal Hermitian self-dual codes in the next section we need the matrices with each row having as many non-zero entries as possible. For that, we propose the following algorithm:
  Algorithm 1
 

Input: : positive integers
Output: A list of unitary matrices

  1. transposition;

  2. cycle of length ;

  3. linear map in Eq. (3);

  4. linear map in Eq. (3);

  5. ;

  6. ;

  7. ;

  8. ;

  9. for to do

  10. for to do

  11. for to do

  12. for to do

  13. ;

  14. Append;

  15. end for;

  16. end for;

  17. end for;

  18. end for;

  19. return ;

 

Iii Construction of Hermitian self-dual codes

In this section, we introduce some constructions of Hermitian self-dual codes based on the elements in the unitary group.

First let us recall the classical constructions of Hermitian self-dual codes provided by Gaborit [11], which are known to be the pure and bordered quadratic double circulant construction whose generator matrices of the code are of the following forms

and

respectively, where and is a circulant matrix indexed by quadratic residues in .

There is also a recursive construction of Hermitian self-dual codes given by Kim et al. [25] as follows.

Proposition 2

([building-up][25]) Let be in such that in . Let be a generator matrix (not necessarily in standard form) of a self-dual code over of length , where are the rows of the matrices , for . Let be a vector in with in . Suppose that for . Then the following matrix:

generates a Hermitian self-dual code over of length

It should be noted that if is a primitive root of and , then we have Thus we derive the construction of Hermitian self-dual codes over as follows.

Proposition 3

Let with being a prime. Let and fix such that Then the matrix of the following form:

(6)

generates a ary self-dual code.

Proof: First note that if is in then so is . Let be the th row of . It follows that for which means that the code having as its generator matrix is Hermitian self-orthogonal. Since the dimension of the code is equal to the row rank of , the result follows.  

Similar to Construction (6), we have the following.

Proposition 4

Let with being a prime. Let and fix such that Then the matrix of the following form:

(7)

generates a ary Hermitian self-dual code.

Proof: Let be the th row of . It follows that for which means that the code having as its generator matrix is Hermitian self-orthogonal. Since the dimension of the code is equal to , the result follows.  

In the rest of the paper, denotes the matrix with all entries equal to

Proposition 5

Let and Fix such that Then for any , a code with the following generator matrix is a ary Hermitian self-dual code:

(8)

Proof: Let be the th row of . It follows that for which means that the code having as its generator matrix is Hermitian self-orthogonal. Since the dimension of the code is equal to the row rank of , the result follows.  

Similarly we have the following construction.

Proposition 6

Let and Fix such that Then for any , a code with the following generator matrix is a ary Hermitian self-dual code:

(9)
Lemma 2

Let and such that Let and denote its th row. Assume there exist in satisfying

(10)

Then the code with the following generator matrix

(11)

is a ary Hermitian self-orthogonal code. Moreover if and then generates a ary Hermitian self-dual code.

Proof: Let be the code generated by . For , let be the th row of A simple calculation implies that

For to be Hermitian self-orthogonal, we have to take and the system (10). Now since is invertible, the row rank of the matrix is at least . If and then by applying elementary row operations on the last rows of followed by swapping the first column and the th column, it is easy to see that the row rank of is exactly and thus the code is ary Hermitian self-dual.  
We deduce the constructions of ary Hermitian self-dual codes from the matrix in Eq. (11) as follows.

Theorem 1

Assume that Then

  1. for the matrix generates a ary Hermitian self-dual code.

  2. for the matrix generates a ary Hermitian self-dual code.

  3. for the matrix generates a ary Hermitian self-dual code.

Proof: It is obvious that and satisfy the first equation of system (10).

  1. Plugging in system (10) and letting be arbitrary, we obtain two equations in two varaibles having the solutions in the desired form.

  2. Plugging in system (10) and letting , we obtain two equations in two varaibles having the solutions in the desired form. If , then by raising both sides to the power , we get that which is a contradiction.

  3. Plugging in system (10) and letting be arbitrary, we obtain two equations in two varaibles having the solutions in the desired form.

 

Remark 2
  1. It should be noted that if and then the code with the generator matrix (11) is still ary Hermitian self-dual but it has minimum distance at most .

  2. For 1) and 3), the parameters of the constructed codes are the same for any so we can choose

  3. Taking in Theorem 1 1), we can express Equation (11) as

    By applying elementary row operations on , we obtain

    which can be viewed as a generalized construction of the quadratic double circulant construction.

Lemma 3

Assume that Let denote the th row of of Eq. (11). If is in span then and for some non-zero

Proof: Assume that for some Considering the first equations defined by this system, we get that Thus and the result follows.

 

Theorem 2

Assume that Then

  1. for the matrix generates a ary Hermitian self-dual code.

  2. for the matrix generates a ary Hermitian self-dual code.

  3. for the matrix generates a ary Hermitian self-dual code.

  4. for the matrix generates a Hermitian self-dual code.

Proof: It is obvious that and satisfy the first equation of system (10).

  1. Plugging in system (10) and letting be arbitrary, we obtain two equations in two varaibles having the solutions in the desired form. From Lemma 3, if is in then we can also write and . Since , we get which is a contradiction. Hence the code dimension is and the result follows.

  2. The result follows from the same computation and reasoning as 1).

  3. Plugging in system (10) and letting , we obtain two equations in two varaibles having the solutions in the desired form. If , then by raising both sides to the power , we get that that is , which is a contradiction. We now prove that the code dimension is . From Lemma 3, if is in then we can also write and . Since , we get and thus which is a contradiction.

  4. Plugging in system (10) and letting be arbitrary, we obtain two equations in two varaibles having the solutions in the desired form. It remains to prove that the code dimension is . First note that if then by raising both sides to power , we get and hence , which is a contradiction to the hypothesis. If then from , we get

    and by raising both sides to the power , we also obtain

    Matching the two equations together gives and hence which means that Since we get and thus Now since for some positive integer , we obtain and it implies that which is a constradiction. Hence and the result follows by Lemma 3.

 

Remark 3
  1. When , the constructed codes in Theorem 2 2) are not ary Hermitian self-dual since the first row is in the spanned space of the other rows, more precisely .

  2. When , the construction in Theorem 2 3) still make sense and by exchanging the first and th columns, it is equivalent to that in Theorem 1 3).

Corollary 1

Assume that and Then for , the matrix generates a ary Hermitian self-dual code.

Proof: The result follows by plugging in Theorem 2 1).  

Corollary 2

Assume that and Then for , the matrix generates a ary Hermitian self-dual code.

Proof: The result follows by plugging in Theorem 2 1).  

Similar to Lemma 2 we have the following.

Lemma 4

Let and such that Let and denote its th row. Assume there exist satisfying

(12)

Then the code with the following generator matrix

(13)

is a ary Hermitian self-orthogonal code. Moreover if and then generates a ary Hermitian self-dual code.

Proof: Let be the code generated by . For , let be the th row of A simple calculation implies that

The rest follows from the same reasoning as that in Lemma 2  

We deduce the constructions of ary Hermitian self-dual codes from the matrix in Eq. (13) as follows.

Theorem 3

Assume that Then

  1. for the matrix generates a ary Hermitian self-dual code.

  2. for the matrix generates a ary Hermitian self-dual code.

  3. for the matrix generates a ary Hermitian self-dual code.

Proof: The proof follows from the same reasoning as that in Theorem 1.  

Lemma 5

Assume that Let denote the th row of of Eq. (11). If is in span then and for some non-zero

Proof: Assume that for some