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, 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. This class of codes was introduced by Massey 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 sidechannel attacks (SCAs) and fault injection attacks (FIAs) was investigated by Carlet et al. [10, 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 selforthogonal (resp. dual containing) codes. In [17, 18], Calderbank et al. presented an effective mathematical method to construct good stabilizer quantum codes from classical selforthogonal 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 selforthogonality.
In [19], Brun et al. introduced entanglementassisted quantum errorcorrecting codes (EAQECCs), which include the standard stabilizer quantum codes as a special case. They showed that if preshared entanglement between the encoder and decoder is available, the EAQECCs can be constructed via classical linear codes without selforthogonality. 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 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 first give some new classes of MDS codes with Euclidean hulls of arbitrary dimensions which 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 byLet be an linear code of length , the Euclidean dual code of is defined as
Similarly, for any two vectors , the Hermitian inner product of u and v is defined as
Let be an linear code of length . We can similarly define the Hermitian dual code of as follows:
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 selforthogonal (resp. Hermitian selforthogonal) code.
Let be distinct elements of and be nonzero elements of . Put and . The generalized ReedSolomon (GRS for short) code over associated to a and v is defined as follows:
It is well known that the code is an MDS code.
The extended GRS code associated to a and v is defined by
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
(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
Lemma 2.
([15, Lemma III.2]) A codeword of is contained in if and only if there exists a polynomial with , such that
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
Lemma 4.
([24, Lemma 7]) A codeword of is contained in if and only if there exists a polynomial with , such that
Lemmas 14 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 . For and , let be an subspace of (or in the proof of Theorem 4) of dimension . Choose (or ). Label the elements of as . For , define
Let and
(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
In particular, let , then
Proof.
Suppose , for some . Then
Note that
and for ,
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 , there exists a ary MDS code with .
 (ii)

If is even, then for any and , there exists a ary MDS code with .
 (iii)
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
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
Since , we have
(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
for any . It follows from and that . Thus
for some with . It deduces that .
Conversely, let be a polynomial of form , where and . We set , then and
By Lemma 1,
Therefore , hence .
(ii) Denote . Let v be defined as 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
(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
for any . It follows from and that . Thus
for some with . It deduces that .
Conversely, let be a polynomial of form , where and . We take , then and
By Lemma 2,
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
By Lemma 6, there exists a monic polynomial of such that
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
(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
for any . It follows from and that . Thus
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
By Lemma 2,
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 a
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