Improved Singleton bound on insertion-deletion codes and optimal constructions

05/05/2021
by   Bocong Chen, et al.
0

Insertion-deletion codes (insdel codes for short) play an important role in synchronization error correction. The higher the minimum insdel distance, the more insdel errors the code can correct. Haeupler and Shahrasbi established the Singleton bound for insdel codes: the minimum insdel distance of any [n,k] linear code over 𝔽_q satisfies d≤2n-2k+2. There have been some constructions of insdel codes through Reed-Solomon codes with high capabilities, but none has come close to this bound. Recently, Do Duc et al. showed that the minimum insdel distance of any [n,k] Reed-Solomon code is no more than 2n-2k if q is large enough compared to the code length n; optimal codes that meet the new bound were also constructed explicitly. The contribution of this paper is twofold. We first show that the minimum insdel distance of any [n,k] linear code over 𝔽_q satisfies d≤2n-2k if n>k>1. This result improves and generalizes the previously known results in the literature. We then give a sufficient condition under which the minimum insdel distance of a two-dimensional Reed-Solomon code of length n over 𝔽_q is exactly equal to 2n-4. As a consequence, we show that the sufficient condition is not hard to achieve; we explicitly construct an infinite family of optimal two-dimensional Reed-Somolom codes meeting the bound.

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1 Introduction

Insertion-deletion codes (insdel codes for short) are designed to protect against synchronization errors [5], [6] in communication systems caused by the loss of positional information of the message. Insdel codes have found applications in many interesting fields such as DNA storage, DNA analysis [7], [17], language processing [2], [12] and race-track memory error correction [3].

The insdel distance between two vectors is defined as the smallest number of insertions and deletions needed to transform one codeword into another. The minimum insdel distance of a code is defined in the natural way: the minimum insdel distance among all its distinct codewords. Like the classical linear codes with respect to the Hamming distance, the minimum insdel distance of an insdel code is an important parameter, which shows its insdel error-correcting capability. The higher the minimum insdel distance, the more insdel errors the code can correct. The study of insdel codes can be date back to the 1960s

[15]. Insdel codes with small code lengths were constructed explicitly by using various mathematical methods, e.g., see [1], [10], [18]. Sloane [13] constructed a family of codes capable of correcting single deletion.

For a fixed code length, it would certainly be nice if both the code size (which is a measure of the efficiency of the code) and the minimum insdel distance could be as large as possible. However, as in the Hamming metric case, these two parameters are restricted each other for any fixed code length. The Singleton bound for insdel codes says that the minimum insdel distance of any linear code over satisfies , see [5]. It is, therefore, natural to consider the problem of constructing insdel linear codes achieving the Singleton bound with equality. Unfortunately, there have been few constructions of Reed-Solomon codes with high capabilities, but none of them meets or comes close to this bound. For example, Wang et al. [16] constructed a class of Reed-Solomon codes with code length , dimension and minimum insdel distance at most ; Tonien [14] et al. constructed a class of generalized Reed-Solomon codes of length and dimension with deletion error-correcting capability of up to . These led to investigate whether the Singleton bound is a tight upper bound for the minimum insdel distance of Reed-Solomon codes. In 2007, McAven et al. [11] showed that Reed-Solomon codes of length and dimension over prime fields can never meet the Singleton bound. Recently, Do Duc et al. [4] improved the result by showing that when the field size is sufficiently large compared to the code length, the Singleton bound cannot be achieved by Reed-Solomon codes; more explicitly, it was shown that the minimum insdel distance of a -dimensional Reed-Solomon code is at most if the code length satisfies and ([4, Theorem 1]); optimal codes that meet the new bound were also constructed explicitly [4, Theorems 2 and 3]. Very recently, Liu et al. in [8] established a set of sufficient conditions for two-dimensional insdel Reed-Solomon codes to have optimal asymptotic error-correcting capabilities.

The aforementioned works lead us to the study of Singleton-type bounds for the minimum insdel distance of general linear codes and optimal constructions of such codes. The contribution of this paper is twofold. We first show that the minimum insdel distance of any linear code over satisfies if . This result improves and generalizes [4, Theorem 1] in two directions: First, our result holds true for general linear codes, not just Reed-Solomon codes; second, we do not require that . More precisely, we obtain the following result.

Theorem A  Suppose is an linear code over with . Then the minimum insdel distance of is at most , i.e., .

We then give a sufficient condition under which the minimum insdel distance of a two-dimensional Reed-Solomon code of length over is exactly equal to . Our approach and conclusion are quite different from those given in [4, Theorems 2 and 3]: the proofs for [4, Theorems 2 and 3] are long and technical, and [4, Theorems 3] requires conditions on the divisors of and some related values; our methods are more natural and direct, and our result mainly concerns the order of the finite field. Our result is stated below.

Theorem B  Let be a finite field with elements, where is a prime number and is a positive integer. Suppose is a primitive element of , i.e., the order of in the multiplicative group of is equal to . Let

be a two-dimensional Reed-Solomon code of length over . We assume that . If

  • and

  • the number of elements of is equal to ,

then the minimum insdel distance of is equal to . In other words, is optimal in the sense that it meets the bound obtained in Theorem A.

As a consequence, we show that the conditions and listed in Theorem B are not hard to achieve; we explicitly construct an infinite family of optimal two-dimensional Reed-Somolom codes meeting the bound, as we show below.

Corollary C  Let be a prime number and let be a positive integer. Let for satisfying . Let be a primitive element in the finite field . Then

is a two-dimensional Reed-Solomon code of length over whose minimum insdel distance is equal to

This paper is organized as follows. In Section , we recall some definitions and basic results about general linear codes, insdel codes and Reed-Solomon codes. In Section , we give the proof for Theorem A, and in Section , the proofs for Theorem B and Corollary C are presented. We conclude this paper with remarks on possible future works in Section .

2 Preliminaries

Let be a finite field with elements and let be the set of all vectors of length over . A subspace of over is called a linear code of length over . The Hamming distance between two vectors , which is defined to be the number of coordinates in which and differ, is denoted by . The minimum Hamming distance of a code is the smallest Hamming distance among all pairs of distinct codewords of . The Hamming weight of a vector is the number of nonzero coordinates in . It is well known that if is a linear code, then the minimum Hamming distance is the same as the minimum Hamming weight of the nonzero codewords of . A linear code of length , dimension and minimum Hamming distance over is often called a -ary code or, if is clear from the context, an code. It is well known that an linear code over must obey the Singleton bound, i.e., the code length , dimension and minimum Hamming distance satisfy

The linear codes over meeting the Singleton bound are called maximum distance separable code (MDS code for short).

In this paper, for linear codes over , we mainly consider the insdel distance used in high insertion and deletion noise regime. We restate this definition as follows.

Definition 2.1.

For two vectors , the insdel distance between and is the minimum number of insertions and deletions which are needed to transform into . It can be verified that is indeed a metric on .

It has been shown that the insdel distance between any two vectors can be characterized via their longest common subsequences.

Lemma 2.2.

[4, Lemma 1] Let . Then we have

where denotes the length of a longest common subsequence of and .

Lemma 2.2 is useful in calculating the insdel distance of two vectors in .

Similar to the definition of minimum Hamming distance of linear codes over , we give the definition of minimum insdel distance of a linear code over below, which is one of the most important parameters as it indicates the insdel error-correcting capability.

Definition 2.3.

An insdel linear code of length is a linear subspace of with minimum insdel distance being defined as

An linear code over of length , dimension and minimum insdel distance is called an insdel linear code over . As we mentioned in the first section, an insdel linear code must obey the following Singleton-type bound.

Proposition 2.4.

(Singleton Bound [5]) Let be an insdel linear code over . Then

In the rest of this section we give the definition and some basic facts about Reed-Solomon codes. Let be two positive integers. Let be a finite field with elements and choose distinct elements of . Denote by the set of polynomials in of degree less that . For , the Reed-Solomon code of length and dimension with code locators is defined as

Then is an linear code over with length . In particular, Reed-Solomon codes are MDS codes.

3 Proof of Theorem A

Let be a linear code of length over . As before, the minimum Hamming distance of the linear code is denoted by ; the minimum insdel distance of is denoted by . For two typical codewords of , the Hamming (resp. insdel) distance between and is denoted by (resp. ). It follows from Lemma 2.2 that the insdel distance between and is less than or equal to , i.e., . In order to prove Theorem A, we first need to improve this upper bound, as we show below.

Lemma 3.1.

Let and be two vectors of length over . Then we have

Proof.

Let denote the length of a longest common subsequence of and . Observe that and then by Lemma 2.2 we immediately have

The lemma is proved. ∎

We have shown that the insdel distance of arbitrary two vectors in is at most twice of their Hamming distance. We next show that the same conclusion holds for the minimum insdel distance and the minimum Hamming distance of any linear code.

Lemma 3.2.

Let be a non-zero linear code of length over . We then have

Proof.

Choose two distinct codewords and of such that . Recall that . Thus by Lemma 3.1 we have

We are done. ∎

Remark 3.3.

Note, by the classical Singleton bound of an linear code , that . We therefore conclude from Lemma 3.2 that the minimum insdel distance of an linear code must be less than or equal to , i.e.,

This Singleton bound for an insdel code was exhibited in [5]. We also note by Lemma 3.2 that if the minimum Hamming distance of is at most , then , proving Theorem A in this special case. By virtue of this fact, for the goal of completing the proof Theorem A, we only need to restrict ourself to the case where is an MDS code.

The following lemma gives the desired result, which is a crucial step in the process of proving our Theorem A.

Lemma 3.4.

Let be a linear MDS code over with . Then

Proof.

We use a characterization of MDS codes to complete the proof: An linear code over is MDS if and only if has a minimum weight codeword in any coordinates (see [9, Chap. 11, Theorem 4]). Now is an MDS code over with , which gives that has a minimum weight codeword in any coordinates. First we choose the last coordinates to be non-zero, and we suppose further that the th coordinate is equal to , i.e.,

where denotes some non-zero elements of . It follows that suitable non-zero elements of can be found such that is a codeword of . Likewise, suitable non-zero elements of can be found such that

is also a codeword of . Now we have two distinct codewords and . Thus the length of a longest common subsequence of and satisfies . Therefore, by Lemma 2.2 we get that

We are done. ∎

By Remark 3.3 and Lemma 3.4, we immediately arrive at Theorem A, which says that the minimum insdel distance of any linear code with is at most .

4 Proofs of Theorem B and Corollary C

The primary goal of this section is to present a proof for Theorem B, which gives a sufficient condition to guarantee a two-dimensional Reed-Solomon code of length over to have minimum insdel distance . Such codes are optimal in the sense that they meet the upper bound obtained in Theorem A.

We first fix some notation. Let be a finite field with elements, where is a prime number and is a positive integer. Let be the multiplicative group of , and let be a primitive element of , i.e., generates the cyclic group . Then is the degree of the minimal polynomial of over the prime field .

Let be integers such that , where . Let be a two-dimensional Reed-Solomon code of length over with code locators , i.e.,

(4.1)

Our goal is, therefore, converting to find suitable numbers such that has minimum insdel distance . For this purpose, let

then

We are now in a position to prove Theorem B.

Proof.

Suppose is a two-dimensional Reed-Solomon code of length over as given in (4.1). According to Theorem A we have that . Thus it remains to show that . To this end, by virtue of Lemma 2.2, it is enough to prove the following claim:

Claim: for any distinct codewords , where is the length of a longest common subsequence of and .

Now we consider cases separately to investigate the number .

Case : Either or is the zero vector. It is trivial to see that because is an MDS code with parameters .

Henceforth, we can assume that both and are non-zero.

Case : Assume that

where are elements of with . Then it is easy to see that .

Case : Assume that

where . Suppose otherwise that there exist three integers satisfying such that

This gives

Thus

Recall that , and for . We then have

This contradicts to the condition in the theorem. It follows that .

Case : Assume that

where . Suppose otherwise that there exist three integers satisfying such that

This leads to

Thus

which gives

This contradicts to the condition in the theorem again. It follows that .

Case : Assume that

where with . Suppose otherwise that there exist six integers satisfying such that

This gives

which implies that

This is equivalent to

Thus we have

This is a contradiction again. It follows that .

Case : Assume that

where . Suppose otherwise that there exist six integers satisfying such that

In the matrix version, that is equivalent to saying that

Since

we have that the determinant of the coefficient matrix is zero, i.e.,

Now suppose is an indeterminate over the finite field . Then we have the polynomial

It is clear that . On the other hand, we can expand the determinant to have

Observing that and , we have

Thus the term of the minimum degree in the polynomial is or . As , otherwise we would have , a contradiction, we conclude that .

Since is a non-zero polynomial, and is the degree of the minimal polynomial of over , we obtain that . Thus it is easy to see that the degree of satisfies

The last inequality is from our condition . This is a contradiction, and we conclude the proof of .

Case : Assume that

where . Suppose otherwise that there exist six integers satisfying such that

We then have

Since

we have that

Using the same discussion as in the Case , we get a contradiction and conclude the proof of .

Based on the above cases, we have established the claim. Then by Lemma 2.2 we obtain , which forces by Theorem A. This completes the proof. ∎

With Theorem B, we can prove Corollary C easily, which generates an infinite family of optimal two-dimensional Reed-Somolom codes meeting the bound in Theorem A.

Proof.

It is enough to check the conditions (1) and (2) in Theorem B are all satisfied. Condition (1) in Theorem B is satisfied since . Next, let us compute . Suppose that

If , then without loss of generality, assume that . This gives , a contradiction, which shows that and . Therefore . We have shown that Condition (2) in Theorem B is satisfied. By Theorem B, we conclude that the code is a two-dimensional Reed-Solomon code of length with minimum insdel distance . ∎

5 Conclusion and future work

In this paper, we showed that if is an linear code over with , then the minimum insdel distance of is at most (see Theorem A). This result significantly improves the previously known results in [4] and [5] as we mentioned in the Introduction section. We gave a sufficient condition under which a two-dimensional Reed-Solomon code of length over has minimum insdel distance (see Theorem B); as a corollary, we showed that the conditions listed in Theorem B are easy to achieve (see Corollary C). Consequently, we have explicitly constructed an infinite family of optimal two-dimensional Reed-Somolom codes meeting the bound in Theorem A. Comparing with [4], our methods are more direct and easy to understand.

A possible direction for future work is to find non-MDS codes that meet the bound exhibited in Theorem A; if this can be done, it may lead us to know more about insertion-deletion metric. Apart from this problem, there could be many other interesting problems associated with insertion-deletion codes. For instance, it would be interesting to establish other bounds with respect to the insertion-deletion metric and give some optimal constructions.

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