# Euclidean and Hermitian Hulls of MDS Codes and Their Applications to EAQECCs

In this paper, we construct several classes of maximum distance separable (MDS) codes via generalized Reed-Solomon (GRS) codes and extended GRS codes, which we can determine the dimensions of their Euclidean hulls or Hermitian hulls. It turns out that the dimensions of Euclidean hulls or Hermitian hulls of the codes in our constructions can take all or almost all possible values. As a consequence, we can apply our results to entanglement-assisted quantum error-correcting codes (EAQECCs) and obtain several families of MDS EAQECCs with flexible parameters. The required number of maximally entangled states of these MDS EAQECCs can take all or almost all possible values. In particular, all possible parameters for the q-ary MDS EAQECCs of length n≤ q are completely determined and several new classes of q-ary MDS EAQECCs of length n>q are also obtained.

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## 1 Introduction

Let be a linear code over a finite field, and let be the dual code of with respect to some inner product, such as Euclidean inner product and Hermitian inner product. The hull of is just defined as the intersection . Some research topics in coding theory are closely related to the properties of the hull of a linear code. One interesting problem in coding theory is that to decide whether two matrices generate equivalent linear codes and compute the permutation of two given equivalent linear codes ([1, 2]). In [3, 4, 5, 6], the authors provided some algorithms for these computations whose complexity is determined by the dimension of the Euclidean hull of codes. Some properties of the hulls of cyclic codes and negacyclic codes were also studied in [7] and [8].

It is worth mentioning that two special cases of the hulls of linear codes are of much interest. One is that , in which is called a linear complementary dual (LCD) code. In [9], Massey first introduced this class of codes and proved that there exist asymptotically good LCD codes. A practical application of binary LCDs against side-channel attacks (SCAs) and fault injection attacks (FIAs) was investigated by Carlet et al. in [10] and [11]. The study of LCD codes is thus becoming a hot research topic in coding theory ([12, 13, 14, 15, 16]). A surprising result was given in [16], which proved that any linear code over is equivalent to a Euclidean LCD code and any linear code over is equivalent to a Hermitian LCD code. The other case is that (resp. ). Such codes are called self-orthogonal (resp. dual containing) codes. In [17, 18], Calderbank et al. presented an effective mathematical method to construct good stabilizer quantum codes from classical self-orthogonal codes (or dual containing codes) over finite fields. Since then, several families of stabilizer quantum codes have been constructed by classical linear codes with certain self-orthogonality.

In [19], Brun et al. introduced entanglement-assisted quantum error-correcting codes (EAQECCs), which include the standard stabilizer quantum codes as a special case. They showed that if pre-shared entanglement between the encoder and decoder is available, the EAQECCs can be constructed via classical linear codes without self-orthogonality. Moreover, an EAQECC is MDS if and only if the corresponding classical linear code is MDS. However, it is not easy to determine the number of shared pairs that required to construct an EAQECC. Several classes of MDS EAQECCs had been constructed with some specific values of the numbers of shared pairs ([20, 21]). Guenda et al. [22] provided the relation between this number and the dimension of the hull of classical linear codes. Therefore, it is important to study the hull of linear codes, in particular for MDS codes. Very recently, Luo et al. [23] presented several classes of GRS and extended GRS codes with Euclidean hulls of arbitrary dimensions and constructed some familes of MDS EAQECCs with flexible parameters. However, the lengths of these -ary MDS EAQECCs are bounded by in their paper. In addition, they only consider the linear codes with Euclidean inner product and the extended GRS codes of specific length ( is the size of the base field).

In this paper, we construct several MDS codes by utilizing GRS codes and extended GRS codes, and determine the dimensions of their Euclidean or Hermitian hulls. More precisely, we firstly give some new classes of MDS codes with Euclidean hulls of arbitrary dimensions whose parameters are not covered in [23]. Secondly, several new classes of MDS codes with Hermitian hulls of arbitrary dimensions are presented. Finally, we apply these results to construct new MDS EAQECCs. In particular, some of these -ary MDS EAQECCs have lengths larger than and the required number of maximally entangled states can take all or almost all possible values. Furthermore, all possible parameters for the -ary MDS EAQECCs of length are completely determined.

The rest of this paper is organized as follows. In Section 2, we briefly recall some basic notions and properties of GRS codes and extended GRS codes. In Section 3, we present our constructions of MDS codes with Euclidean or Hermitian hulls of arbitrary dimensions. Several classes of MDS EAQECCs are obtained in Section 4. We conclude this paper in Section 5.

## 2 Preliminaries

In this section, we briefly recall some basic notions and properties of GRS codes and extended GRS codes.

Throughout this paper, we always assume that is a prime and , where is positive integer. Let be the finite field with elements and

. For any two vectors

the Euclidean inner product u and v is defined by

 ⟨u,v⟩E:=n∑i=1uivi.

Let be an -linear code of length , the Euclidean dual code of is defined as

 C⊥E:={u∈Fnq:⟨u,v⟩E=0 for all v∈C}.

Similarly, for any two vectors , the Hermitian inner product of u and v is defined as

 ⟨u,v⟩H:=n∑i=1uivqi.

Let be an -linear code of length . We can similarly define the Hermitian dual code of as follows:

 C⊥H:={u∈Fnq2:⟨% u,v⟩H=0 for all v∈C}.

It is worth mentioning that the base field should be when we consider the Hermitian case in this paper.

The Euclidean hull (resp. Hermitian hull) of is just the intersection (resp. ), which we denote by (resp. ). It is obvious that (resp. ). If (resp. ), is called a Euclidean LCD (resp. Hermitian LCD) code. If (resp. ), is called a self-orthogonal (resp. Hermitian self-orthogonal) code.

Let be distinct elements of and be nonzero elements of . Put and . The generalized Reed-Solomon (GRS for short) code over associated to a and v is defined as follows:

 GRSk(a,v):={(v1f(a1),…,vnf(an)):f(x)∈Fq[x],deg(f(x))≤k−1}.

It is well known that the code is an -MDS code.

The extended GRS code associated to a and v is defined by

 GRSk(a,v,∞) := {(v1f(a1),…,vnf(an),fk−1) :f(x)∈Fq[x],deg(f(x))≤k−1},

where stands for the coefficient of in . It is easy to show that is an -MDS code (see [25, Theorem 5.3.4]). For , we denote

 ui:=∏1≤j≤n,j≠i(ai−aj)−1, (1)

which will be used frequently in this paper.

In [15], the authors presented a sufficient and necessary condition under which a codeword c of (resp. ) is contained in its dual code (resp. ).

###### Lemma 1.

([15, Lemma III.1]) A codeword of is contained in if and only if there exists a polynomial with , such that

 (v21f(a1),v22f(a2),…,v2nf(an))=(u1g(a1),u2g(a2),…,ung(an)).
###### Lemma 2.

([15, Lemma III.2]) A codeword of is contained in if and only if there exists a polynomial with , such that

 (v21f(a1),v22f(a2),…,v2nf(an),fk−1) = (u1g(a1),u2g(a2), …,ung(an),−gn−k).
###### Remark 1.

Indeed, [15, Lemma III.2] only considered the case of . It can similarly prove that the lemma holds for general .

Similar results for the Hermitian case were obtained in [24].

###### Lemma 3.

([24, Lemma 6]) A codeword of is contained in if and only if there exists a polynomial with , such that

 (vq+11fq(a1),vq+12fq(a2),…,vq+1nfq(an))=(u1g(a1),u2g(a2),…,ung(an)).
###### Lemma 4.

([24, Lemma 7]) A codeword of is contained in if and only if there exists a polynomial with , such that

 (vq+11fq(a1),vq+12fq(a2),…,vq+1nfq(an),fqk−1) = (u1g(a1),u2g(a2), …,ung(an),−gn−k).

Lemmas 1-4 will play important roles in calculating the dimensions of hulls of the MDS codes constructed in Section 3.

## 3 Constructions

In this section, we will provide several families of GRS codes and extended GRS codes with Euclidean hulls or Hermitian hulls of arbitrary dimensions. The main idea of our constructions is to choose suitable distinct elements (or ) such that each value of defined by Eq. (1) can be easily calculated.

### 3.1 MDS Codes with Euclidean Hulls of Arbitrary Dimensions

In this subsection, we will provide some constructions of MDS codes with Euclidean hulls of arbitrary dimensions. Since , we always assume that the dimension is less than or equal to half of the code length in our constructions.

The first construction is based on an additive subgroup of and its cosets. Let and , where . Then can be seen as a linear space over of dimension . Suppose and , let be an -subspace of (or in the proof of Theorem 4) of dimension . Choose (or ). Label the elements of as . For , define

 Hj:=H+βjη:={h+βjη∣h∈H}.

Let and

 t⋃j=1Hj:={a1,a2,…,an}. (2)

For , is defined as in (1). Similar to [24, Lemmas 8 and 9], the value of is given as follows.

###### Lemma 5.

For a given , suppose for some . Then we have

 ui=(∏h∈H,h≠0h−1)(∏g∈H(η−g))1−t⎛⎝∏1≤j≤t,j≠b(βb−βj)−1⎞⎠.

In particular, let , then

 εui∈F∗r.
###### Proof.

Suppose , for some . Then

 ui=∏1≤j≤n,j≠i(ai−aj)−1=∏hb∈Hb,hb≠ai(ai−hb)−1∏1≤j≤t,j≠b⎛⎝∏hj∈Hj(ai−hj)−1⎞⎠.

Note that

 ∏hb∈Hb,hb≠ai(ai−hb)==∏γ∈H,γ≠ξ(ξ+βbη−(γ+βbη))=∏h∈H,h≠0h,

and for ,

 ∏hj∈Hj(ai−hj) = ∏γ∈H(ξ+βbη−(γ+βjη)) = ∏g∈H((βb−βj)η−g) = (βb−βj)∏g∈H(η−g).

The last equality holds since . The lemma is proved. ∎

Before giving our constructions, we need the following simple lemma.

###### Lemma 6.

Let be a finite field and . Then, for any integer , there exists a monic polynomial of degree such that for all .

###### Proof.

For , let ; For , let , where For , the conclusion follows from [24, Lemma 12]. ∎

###### Theorem 1.

Let and , where . Suppose is even. Let , where and .

(i)

For any and , then there exists a -ary -MDS code with .

(ii)

If is even, then for any and , there exists a -ary -MDS code with .

(iii)

If

is odd and

, then for any and , there exists a -ary -MDS code with .

###### Proof.

Let be defined as (2) and be defined as in Lemma 5. Choose with .

(i) Since is even, each element of is a square in . By Lemma 5, there exist such that

 εui=v2i,

for . Denote . Put and . We consider the Euclidean hull of the -MDS code . For any with . By Lemma 1, there exists a polynomial with such that

 (α2v21f(a1),…,α2v2sf(as),v2s+1f(as+1),…,v2nf(an)) =(u1g(a1),…,usg(as),us+1g(as+1),…,ung(an)).

Since , we have

 (α2εu1f(a1),…,α2εusf(as),εus+1f(as+1),…,εunf(an))=(u1g(a1),…,usg(as),us+1g(as+1),…,ung(an)). (3)

From the last coordinates of Eq. (3), we obtain that for any . Since , . Note that and , thus . On the other hand, the first coordinates of Eq. (3) imply that

 α2εuif(ai)=uig(ai)=εuif(ai),

for any . It follows from and that . Thus

 f(x)=h(x)s∏i=1(x−ai),

for some with . It deduces that .

Conversely, let be a polynomial of form , where and . We take , then and

 (α2v21f(a1),…,α2v2sf(as),v2s+1f(as+1),…,v2nf(an)) =(u1g(a1),…,usg(as),us+1g(as+1),…,ung(an)).

By Lemma 1,

 (αv1f(a1),…,αvsf(as),vs+1f(as+1),…,vnf(an))∈HullE(C).

Therefore , hence .

(ii) Denote . Let v be defined as in the proof of part (i). We consider the Euclidean hull of the -MDS code . For any with . By Lemma 2, there exists a polynomial with such that

 (α2εu1f(a1),…,α2εusf(as),εus+1f(as+1),…,εunf(an),fk−1)=(u1g(a1),…,usg(as),us+1g(as+1),…,ung(an),−gn−k). (4)

From Eq. (4), we can similarly deduce that and . If , then , i.e., which contradicts to the assumption that is even. Thus and . On the other hand, the first coordinates of Eq. (4) imply that

 α2εuif(ai)=uig(ai)=εuif(ai),

for any . It follows from and that . Thus

 f(x)=h(x)s∏i=1(x−ai),

for some with . It deduces that .

Conversely, let be a polynomial of form , where and . We take , then and

 (α2v21f(a1),…,α2v2sf(as),v2s+1f(as+1),…,v2nf(an),0) =(u1g(a1),…,usg(as),us+1g(as+1),…,ung(an),0).

By Lemma 2,

 (αv1f(a1),…,αvsf(as),vs+1f(as+1),…,vnf(an),0)∈HullE(C).

Therefore , hence .

(iii) We first claim that is a square in . Indeed, if is even, it is done since each element in is a square. Suppose is odd, if , then and . Thus is a square in , where . Note that is odd, thus is odd and hence is a square. Thus the claim holds. By Lemma 5 and the fact that each element of is a square in , there exist such that

 ui=−v2i, for all 1≤i≤n.

By Lemma 6, there exists a monic polynomial with such that

 π(ai)≠0, for all 1≤i≤n.

Denote . Put and , where . We consider the Euclidean hull of the -MDS code .

For any with . By Lemma 2 and , there exists a polynomial with such that

 −(α2u1π21f(a1),…,α2usπ2sf(as),us+1π2s+1f(as+1),…,unπ2nf(an),−fk−1)=(u1g(a1),…,usg(as),us+1g(as+1),…,ung(an),−gn−k). (5)

From the -th to -th coordinates of Eq. (5), we obtain that for any . Since , . Note that and , thus . Note that , hence . On the other hand, the first coordinates of Eq. (5) imply that

 α2uiπ2(ai)f(ai)=−uig(ai)=uiπ2(ai)f(ai),

for any . It follows from and that . Thus

 f(x)=h(x)s∏i=1(x−ai),

for some of . It deduces that .

Conversely, let be a polynomial of form , where and . We set . Then , and if and only if . Thus . It is directly to verify that

 (α2v21π21f(a1),…,α2v2sπ2sf(as),v2s+1π2s+1f(as+1),…,v2nπ2nf(an),fk−1) =(u1g(a1),…,usg(as),us+1g(as+1),…,ung(an),−gn−k).

By Lemma 2,

 (αv1π1f(a1),…,αvsπsf(as),vs+1πs+1f(as+1),…,vnπnf(an),fk−1)∈HullE(C).

Therefore , hence .

The proof is completed.

In the following theorem, we employ a multiplicative subgroup of and the zero element to construct extended GRS codes with Euclidean hulls of arbitrary dimensions

###### Theorem 2.

Let be a prime power. Assume that is odd, and