Bounds on Subspace Codes Based on Subspaces of Type (𝑚,1) in Singular Linear Space

Abstract

The Sphere-packing bound, Singleton bound, Wang-Xing-Safavi-Naini bound, Johnson bound, and Gilbert-Varshamov bound on the subspace codes based on subspaces of type in singular linear space over finite fields are presented. Then, we prove that codes based on subspaces of type in singular linear space attain the Wang-Xing-Safavi-Naini bound if and only if they are certain Steiner structures in .

1. Introduction

Let be a finite field with elements, where is a power of a prime and is the -dimensional row vector space over , where is a positive integer. The set of all the subspaces with dimension of is called Grassmannian space over , denoted by . The set of all the subspaces of , including and , is called projective space of order over , denoted by . For any subspaces and in , define the distance function between and as The function above is proved a metric (see [1]); thus, can be regarded as a metric space.

A nonempty collection of projective space is called a subspace code and the subspaces in are codewords of code . The minimum distance of a subspace code is A subspace code is denoted by , if the size of codewords is and the minimum distance of is . Furthermore, if , is denoted by .

Subspace code plays an important role in random network coding (see [2, 3]). Koetter et al. [4, 5] defined an operator channel when they studied random network coding. Meanwhile, they showed that the errors and erasures could be corrected by a subspace code over the operator channel if the sum of errors and erasures is less than . These research results motivate many domestic and overseas scholars’ great interest in subspace codes (see [3–8]).

Bounds on subspace codes and optimal subspace codes in projective space are considered in recent years. Koetter and Kschischang [1] provided Sphere-packing bound and Singleton bound on subspace codes in projective space, which are regarded as a counterpart of the classical Sphere-packing bound and Singleton bound [9]. Wang et al. [10] provided the Wang-Xing-Safavi-Naini bound on subspace codes in projective space. Etzion and Vardy [11] provided Johnson bound and Gilbert-Varshamov bound on subspace codes in projective space. Meanwhile, Xia and Fu [12] proved that codes in projective space achieved the Wang-Xing-Safavi-Naini bound if and only if they are certain Steiner structures [13, 14] in .

In this paper, denote by the collection of all the subspaces of type in -dimensional singular linear space , where . For any two subspaces and of type in , the natural distance function between and in is defined as A nonempty collection of is called an code if it has the size of codewords and the minimum distance . Denote by the maximum number of codewords in an code. The goal of this paper is to determine bounds of and construct the corresponding optimal subspace codes.

The Sphere-packing bound, Singleton bound, Wang-Xing-Safavi-Naini bound, Johnson bound, and Gilbert-Varshamov bound on the subspace codes based on subspaces of type in singular linear space over are given for the first time in this paper. We prove that the Steiner structure is an code, which attains the Wang-Xing-Safavi-Naini bound. The study of optimal subspace codes is one of the main research problems in coding theory. The obtained bounds are of vital importance to construct the optimal subspace codes.

The rest of the paper is organized as follows. In Section 2, the relevant concepts of -dimensional singular linear space over are introduced and the anzahl formulas in singular linear space are given. In Section 3, the Sphere-packing bound, Singleton bound, Wang-Xing-Safavi-Naini bound, Johnson bound, and Gilbert-Varshamov bound on the subspace codes based on subspaces of type in singular linear space over are presented. In Section 4, we prove that the Steiner structure is an code, which attains the Wang-Xing-Safavi-Naini bound. That is, the Steiner structure is an optimal subspace code. In Section 5, a conclusion is made for this paper.

2. Preliminaries

Let be a finite field with elements, where is a prime power. is the -dimensional row vector space over , where and are two nonnegative integers. The set of all nonsingular matrices over of the form where and are nonsingular and matrices, respectively, forms a group under matrix multiplication, called the singular general linear group of degree over and denoted by .

We have an action of on defined as follows: The vector space , together with the above group action, is called the -dimensional singular linear space over (see [15]).

Let be the row vector in whose th coordinate is and all other coordinates are . Denote by the -dimensional subspace of generated by . An -dimensional subspace of is called a subspace of type if .

Introduce the anzahl formulas (see [15]) and use the Gaussian coefficient [16] for brevity: By convenience, and whenever and .

Denote the set of all the subspaces of type in by and let It is verified that is nonempty if and only if Moreover, if is nonempty, then

Let be a fixed subspace of type in . Denote by the set of all the subspaces of type contained in . Let It is verified that is nonempty if and only if Moreover, if is nonempty, then

Let be a fixed subspace of type in . Denote by the set of all the subspaces of type containing . Let It is verified that is nonempty if and only if Moreover, if is nonempty, then

3. Bounds on Subspace Codes Based on Subspaces of Type in Singular Linear Space

Denote by the maximum number of codewords in an code based on subspaces of type in singular linear space . By (3), the distance of any two elements in must be an even number; thus, we only need to consider for even .

Definition 1. The sphere of radius centered at a subspace of type in is defined as the set of all subspaces of type in that satisfy . That is,

Theorem 2. The number of subspaces of type in is independent of the choice of and where is a fixed subspace of type in . is the number of elements satisfying is the number of elements satisfying

Proof. Let be a fixed subspace of type in . Let denote the number of elements satisfying . Choose the subspace of type of intersection in ways. Once this is done, the subspace can be extended to a subspace of type in ways. Thus, From this,
Let denote the number of elements satisfying . Choose the subspace of type of intersection in ways. Once this is done, the subspace can be extended to a subspace of type in ways. Thus, From this,
and denote the number of subspaces of type at distance from the fixed subspace of type . Therefore, which is clearly independent of the choice of .

Theorem 3 (Sphere-packing bound). Let , and then

Proof. Let be an code and let . Then the spheres of radius centered at every codeword of disjoint with each other and each of these spheres contains subspaces of type in . Since the total number of subspaces of type in is Then the theorem follows immediately.

Suppose is a collection of subspaces of type in and let be any -dimensional subspace of . Define by replacing each subspace by , where is defined as follows: is replaced by if is a subspace of type in ; otherwise, is replaced by some subspace of type of .

Lemma 4. If is an code with , then is an code with .

Proof. It is sufficient to consider the cardinality and the minimum distance of code . Let and be any two distinct codewords of code and the corresponding codewords and of code are obtained. Obviously and , and then and where the latter inequality follows from the fact that Considering with (33), which implies that .
By , , so and are distinct, which shows that code has the same number of codewords as code .

Theorem 5 (Singleton bound). Consider

Proof. Let be an code. When Lemma 4 is applied times, an code is obtained, where every codeword is of subspace of type in . It is well known that Hence where the inequality follows from the fact that the obtained code cannot have more codewords than the total subspaces of type in . Then the theorem follows.

Theorem 6 (Wang-Xing-Safavi-Naini bound). Consider

Proof. Let be an code. On the one hand, each codeword of contains exactly subspaces of type .
On the other hand, if for any two distinct codewords and of , by (3), a contradiction to the minimum distance of , which implies that a given subspace of type in cannot be contained in two distance codewords of . The total number of subspaces of type in is so .

Theorem 7 (Johnson bound). Consider

Proof. Let be an code, which attains the value of ; that is, . Each codeword (subspace of type ) of contains subspaces of type in . There are totally subspaces of type in . Hence, there exists a subspace of type satifying the fact that the subspace is contained in at least codewords of .
Let , where is a subspace of type in . Define the following set: Clearly consists of the subspaces of type of ; thus, can be regarded as an code with the size and the minimum distance , where .
Let and be any two codewords of and there exist two corresponding codewords and of such that and with and . Noting that where the third equality follows from the fact that contains , hence, By (3) and (49), from which follows. With all the above discussions, The theorem is completed.

Corollary 8. Consider

Proof. When Theorem 7 is applied iteratively times, note the trivial equality

Corollary 9. Consider

Theorem 10 (Johnson bound). Consider

Proof. Let be an code, which attains the value of ; that is, . For each define the following set: Clearly, for each , is an code with the size and the minimum distance and . There are precisely elements containing any given subspace of type in ; hence each codeword (subspace of type ) of is contained in different codes . Consider from which where Thus, there exists at least one such that Obviously, for any , and then the theorem follows immediately.

Theorem 11 (Gilbert-Varshamov bound). If ,  , then where

Proof. Let be an code, which attains the value of ; that is, . There is no subspace of type in satisfying for any . Otherwise, by adding the subspace of type to , a new code is obtained, which is a contradiction to . Therefore, for any , That is, where

4. Steiner Structure

Definition 12. A Steiner structure is a collection of elements from satisfying the fact that each element from is contained in exactly one element of . The subspaces of type in are called blocks of the Steiner structure .

Theorem 13. The total number of blocks in the is

Proof. Each block (subspace of type ) of the Steiner structure contains subspaces of type . By the definition of the Steiner structure , each subspace of type is contained in exactly one block. The total number of subspaces of type in is thus, the total number of blocks of the is

Theorem 14. The Steiner structure is an code with the size and .

Proof. By Definition 12 and Theorem 13, it is sufficient to prove . Firstly, for any two distinct blocks , where the inequality follows from the definition of the Steiner structure in which each subspace of type is contained in exactly one block of the . By (3) and (74), Furthermore, is the minimum distance of the code; then .
Conversely, let be a fixed subspace of type in , and then choose two subspaces and of type in such that By the definition of the Steiner structure , there exist the uniquely corresponding subspaces and of type in the Steiner structure satisfying respectively. Then, from which By (3) and (79), Furthermore, is the minimum distance of the code, and then .
From the above two aspects, .

The sufficient and necessary conditions for an code to attain the Wang-Xing-Safavi-Naini bound in Theorem 6 are presented below.

Theorem 15. An code attains the Wang-Xing-Safavi-Naini bound; that is, if and only if is the Steiner structure .

Proof. From Theorems 6 and 14, if is the Steiner structure , is an code with Obviously such a code attains the Wang-Xing-Safavi-Naini bound.
Conversely, let be an code attaining the Wang-Xing-Safavi-Naini bound; that is, Firstly, any subspace of type cannot be contained in two distinct codewords and of . Otherwise, by (3), a contradiction to the minimum distance of . Each codeword of contains distinct subspaces of type ; thus, all the codewords of contain totally distinct subspaces of type . There are totally distinct subspaces of type in , which implies that each subspace of type is contained in exactly one codeword of .
When regarding all the codewords of as blocks, code is the Steiner structure .

Theorem 15 shows that certain Steiner structure is an optimal code.

Corollary 16. Consider
if and only if the Steiner structure exists.

5. Conclusion

In this paper, a subspace code based on subspaces of type in singular linear space over is constructed and the Sphere-packing bound, Singleton bound, Wang-Xing-Safavi-Naini bound, Johnson bound, and Gilbert-Varshamov bound on code are presented. Meanwhile, we prove that codes based on subspaces of type in singular linear space attain the Wang-Xing-Safavi-Naini bound if and only if they are certain Steiner structures . We could consider whether there exist nontrivial perfect subspace codes in in the following steps.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

This work is supported by the National Natural Science Foundation of China under Grant no. 61179026 and the (Boeing) Technical Challenge Foundation of Civil Aviation University of China under Grant no. 20140159201. The authors would like to thank the referee for his suggestions on simplifying the earlier version of the paper.