Two-sided Galois duals of multi-twisted codes

Galois duals of Multi-twisted (MT) codes are considered in this study. We describe a MT code C\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {C}$$\end{document} as a module over a principal ideal domain. Hence, C\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {C}$$\end{document} has a generator polynomial matrix (GPM) G\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{G}$$\end{document} that satisfies an identical equation. We prove a GPM formula for the Euclidean dual C⊥\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {C}^\perp $$\end{document} using the identical equation of the Hermite normal form of G\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{G}$$\end{document}. Next, we aim to replace the Euclidean dual with the Galois dual. The Galois inner product is an asymmetric form, so we distinguish between the right and left Galois duals. We show that the right and left Galois duals of a MT code are MT as well but with possibly different shift constants. Some interconnected identities for the right and left Galois duals of a linear code are established and we also introduce the two-sided Galois dual. We use a condition that makes the two-sided Galois dual of a MT code MT, then we describe its GPM. Two special cases are also studied, one when the right and left Galois duals trivially intersect and the other when they coincide. A necessary and sufficient condition is established for the equality of the right and left Galois duals of any linear code.


Introduction
Linear codes with a rich algebraic structure are important in real communication systems due to the potential for developing their encoding and decoding algorithms. One of these codes is the class of cyclic codes over finite fields. Cyclic codes over the finite field F q are in one-to-one correspondence with ideals of the polynomial ring F q [x]. The class of cyclic codes has undergone a series of generalizations to of a linear code will differ if we are going to make these vectors to the left or to the right of the Galois inner product. Therefore, it is necessary to differentiate between the right Galois dual and the left Galois dual of a linear code. To study these duals for MT codes, we inspect the application of a finite field automorphism to a MT code. This produces a MT code with the same block lengths but possibly different shift constants. We deduce the reduced GPM of the resulting MT code and the matrix that satisfies its identical equation from their counterparts of the original MT code. We then prove that the right and left Galois duals of a linear code are the images of its Euclidean dual under some automorphisms. Thus the right and left Galois duals of a MT code are also MT. We prove formulas for their shift constants, their reduced GPMs, and the matrices that satisfy their identical equations. To our knowledge, right and left Galois duals have not appeared simultaneously in any previous study. For instance, the Galois dual introduced in [15] coincides with our definition of the right Galois dual, while the Galois dual introduced in [5] coincides with our definition of the left Galois dual. We find it useful to simultaneously include these two distinct Galois duals in our study. This allowed us to prove their interrelated properties, see Theorem 10. Some of these properties generalize the traditional properties of the Euclidean dual of a linear code. For example, the right (respectively, left) Galois dual of the left (respectively, right) Galois dual is the original code.
Another significant advantage of including the right and left Galois duals simultaneously in our study is to inspect the two-sided Galois dual of a MT code, which we define as the intersection of these two duals. For a MT code, although both its right and left Galois duals are MT, its two-sided Galois dual is not necessarily MT. We use a sufficient condition under which the two-sided Galois dual of a MT code is MT as well. Under this condition, we aim to describe a GPM of the two-sided Galois dual. We begin this direction in a more general context in Theorems 11-16. A particular case of these theorems leads to some constraints whose solution produces the reduced GPM of the two-sided Galois dual and the matrix satisfying the identical equation. With the aid of the trace map over a finite field extension, we provide an auxiliary equation that helps in solving these constraints. An illustrative example shows in detail how to find a solution that satisfies these constraints. Furthermore, two remarkable cases of the two-sided Galois dual of a MT code are considered. The first is when the right and left Galois duals are identical. We establish a necessary and sufficient condition on any linear code to have equal right and left Galois duals. The second is when the right and left Galois duals trivially intersect. An application of the latter case is given in Corollary 5, which presents the condition on a MT code equivalent to writing the vector space as a direct sum of the right and left Galois duals of the code.
The remaining sections are organized as follows. Section 2 summarizes some preliminaries to MT codes, their properties, GPMs, identical equations, and reduced forms of their GPMs. In Sect. 3, we present our results regarding the Euclidean duals of MT codes. However, the results for the right, left, and two-sided Galois duals of a MT code are presented in Sect. 4.

The algebraic structure of a MT code
Let F q be the finite field of order q, where q = p e is a prime power. A code C over F q of length n is linear if it is a subspace of F n q , and hence we can define the dimension of C. The Euclidean inner product on F n q is a symmetric bilinear form defined by for any a = (a 0 , a 1 , . . . , a n−1 ) , b = (b 0 , b 1 , . . . , b n−1 ) ∈ F n q . The Euclidean dual C ⊥ of C is defined by If C is linear of length n and dimension k, one can easily show that C ⊥ is linear of dimension n − k and C ⊥ ⊥ = C. A linear code is called cyclic if it is invariant under the cyclic shift of its codewords by one coordinate. That is, C is cyclic if and only if It is convenient to represent the codewords of a cyclic code as polynomials in the quotient ring R = F q [x]/ x n − 1 . Precisely, (c 0 , c 1 , . . . , c n−2 , c n−1 ) ∈ C has the polynomial representation c 0 + c 1 x + · · · + c n−2 x n−2 + c n−1 x n−1 ∈ R. This representation gives cyclic codes the structure of ideals in R. The cyclic shift property of cyclic codes is generalized to constacyclic codes. Let 0 = λ ∈ F q . A linear code C is called constacyclic with a shift constant λ if In polynomial representation, a constacyclic code over F q of length n and shift constant λ is an ideal in the quotient ring R λ = F q [x]/ x n −λ . But any ideal in R λ corresponds to an ideal in F q [x] containing x n − λ. The latter has a unique monic generator polynomial g(x) that satisfies the identical equation a(x)g(x) = x n −λ; this is because F q [x] is a PID. Thus, constacyclic codes over F q of length n and shift constant λ are in one-to-one correspondence with ideals of F q [x] generated by monic divisors of x n − λ. We aim to present analogous correspondence in the class of MT codes.
A linear code C over F q of length n is called -QC if it is invariant under the cyclic shift of its codewords by coordinates. Thus, C is -QC if and only if (c 1 , c 2 , . . . , c n ) ∈ C ⇒ (c n− +1 , . . . , c n , c 1 , c 2 , . . . , c n− ) ∈ C.
The smallest positive integer with this property is called the index of C and denoted by . Indeed, the index divides the code length n and their quotient is called the co- is a codeword for every c ∈ C in the form of (1). QC codes generalize cyclic codes (when = 1) but not constacyclic codes, however QT codes do. For a nonzero λ ∈ F q , a linear code C over F q of length n is called The index of C is the smallest positive integer with this property, while λ is called the shift constant of C. The index of a QT code divides its length and their quotient is the co-index m. Similar to QC codes, a linear code C of length m is ( , λ)-QT if and only if is a codeword for every c ∈ C in the form of (1). Let T ( ,λ) be the automorphism of We view F m q as an F q [x]-module by defining the action of x as the action of T ,λ . Since an ( , λ)-QT code over To exhibit a polynomial representation for QT codes, where a j (x) = a 0, j + a 1, j x + a 2, j x 2 + · · · + a m−1, j x m−1 ∈ R λ for 1 ≤ j ≤ . The polynomial representation of an ( , λ)-QT code C ⊆ F m q is φ (C). Specifically, the codeword given by (1) is represented by the polynomial vector c(x) = c 0,1 + c 1,1 x + c 2,1 x 2 + · · · + c m−1,1 x m−1 , c 0,2 + c 1,2 x + c 2,2 x 2 + · · · + c m−1,2 x m−1 , . . . , Thus, ( , λ)-QT codes are in one-to-one correspondence with the F q [x]-submodules of R λ , and thus are in one-to-one correspondence with the F q [x]-submodules of F q [x] containing the submodule We do not distinguish between representing an ( , λ)-QT code as a T ,λ -invariant subspace of F m q or representing it as an F q [x]-submodule of F q [x] that contains M. MT codes provide an additional generalization of QT codes by generalizing the block lengths of length m into blocks that are not necessarily equal.
From its definition, a MT code C of index is a linear code over F q [x] of length . A generator matrix for C as a linear code over F q [x] is called GPM because its entries are polynomials over F q . Since a GPM is a matrix over the PID F q [x], one might ask for its unique Hermite normal form, which we call the reduced GPM.

Theorem 1
There is a one-to-one correspondence between (λ 1 , λ 2 , . . . , λ )-MT codes over F q of index and block lengths (m 1 , m 2 , . . . , m ) and T -invariant F q -subspaces of F n q , where n = m 1 + m 2 + · · · + m and T is the automorphism of Proof where a j (x) = a 0, j + a 1, j x + · · · + a m j −1, j x m j −1 for 1 ≤ j ≤ . This gives the commutative diagram of F q -vector space isomorphisms Then xφ (a) = φ (T (a)) for any a ∈ F n q , and φ is an Hereinafter, by a MT code we mean a T -invariant subspace of F n q or a submodule of F q [x] that contains M , and the used algebraic structure is determined from the context. On the other hand, the polynomial representation of a MT-code is the corresponding submodule of ⊕ j=1 R m j ,λ j .
Let C be a (λ 1 , λ 2 , . . . , λ )-MT code over F q of index , block lengths (m 1 , m 2 , . . . , m ), and an r ×n generator matrix G that generates C as an F q -subspace of F n q . Let φ be the map defined by (3) and let Reducing this matrix to the Hermite normal form yields the reduced GPM G of C.
In fact, F q [x] and M are free modules of rank over the PID F q [x] and M ⊆ C ⊆ F q [x] , then C has rank . Consequently, the reduced GPM G = g i, j is upper triangular of rank and size × such that, for 1 ≤ i ≤ , It is worth noting that C = ⊕ j=1 C j , where C j is a λ j -constacyclic code of length m j and generator polynomial g j (x) for 1 ≤ j ≤ , if and only if the reduced GPM of C is the diagonal matrix G = diag [g 1 (x) , . . . , g (x)]. The diagonal matrix is the reduced GPM of the -MT code M . But any -MT code C with a GPM G contains M . Then from Theorem 2, there is a matrix A such that Equation (5) is called the identical equation of G. The matrix A plays a fundamental role in constructing a GPM for the Euclidean and Galois duals of a MT code. If A = a i, j is the matrix that satisfies the identical equation of the reduced GPM, then A is upper triangular and for In particular, A and G commute when C is ( , λ)-QT. The following result can be proven in a similar way to Corollary 3.1 in [5].

Theorem 4 Let C be a -MT code over F q of index and block lengths
Example 1 Let C be the (2, 1)-MT code over F 3 of index = 2, block lengths (m 1 , m 2 ) = (20, 40), and the reduced GPM The matrix that satisfies the identical equation of G is From Theorem 4, C has dimension k = (20 − 14) + (40 − 40) = 6. The minimum distance of C is found to be 36. According to [11], C has the best-known parameters [60, 6, 36] for a linear code over F 3 .
From Theorem 1, an -GQC code can be thought of as: 2. An invariant F q -subspace of F n q , where n = j=1 m j , under the automorphism . . , λ ) while 0 = λ j ∈ F q and m i is a positive integer for 1 ≤ j ≤ . We also let G be a GPM for C, and we denote the matrix that satisfies the identical equation of G by A. In the following result, we prove that the Euclidean dual C ⊥ of C is not only linear, but also MT with the same block lengths but possibly different shift constants. However, the main result of this section is to derive a formula for a GPM of C ⊥ . This will be achieved with the aid of the identical equation of G.
where n = j=1 m j and T is the automorphism given by (2) Observe that applying T exactly N times to any a ∈ F n q keeps a unchanged. Thus T N is the identity map on F n q . If we can show that T C ⊥ = C ⊥ , then C ⊥ is -MT. To do this, consider any b ∈ C ⊥ and c ∈ C. Then Now we define some matrices that are jointly related to the matrix that satisfies the identical equation of the reduced GPM of C. Definition 3 For a MT code C, let G = [g i, j ] be the reduced GPM of C and let A = [a i, j ] be the matrix that satisfies the identical equation of G. For 1 ≤ j ≤ , denote the degree of deg g j, j by d j , i.e., d j = deg g j, j .

Let A 1
x be the matrix obtained from A when x is replaced by 1 x . 2. Let A * be the matrix obtained after multiplying the (i, j)-th entry of A 1 x by be the projection homomorphism and let π = ⊕ h=1 π h . View F n q as an F q [x]-module by defining the action of x as the action of T . Define the Let us fix a positive integer j ≤ and argue as in Definition 3. Suppose that the jth column of A is where d j = deg g j, j . The jth row of H is the jth column of A * * and it satisfies and only if a j gives zero inner product with each codeword in C and that is actually what we will show in the next result.

Lemma 1 For any positive integer j
. Then (5), Applying the same argument in the proof of Theorem 3, we get Reducing the (i, j)-th entry of (7) modulo x N − 1 leads to In fact, G is the reduced GPM of C. (8) can be replaced by Thus, in all cases, a h, j in (8) can be replaced by For any integer 0 ≤ ν ≤ N − 1, the sum of the coefficients of x N −ν−1 and x 2N −ν−1 in (9) is zero. What this shows is that the inner product of a j [see Eq. (6)] and

Lemma 2 The matrix H is a GPM of a -MT code.
Proof Our aim is to prove that x as matrices over the ring F q x, 1 x . From Definition 3, Thus, Thus, the diagonal elements of A * have no negative powers of x. Again from Definition 3, there is a strictly upper triangular matrix S such that Therefore, Note that U is an upper triangular invertible matrix because its determinant is a unit in F q x, 1 x . Then, Then B is lower triangular such that The diagonal elements of B and H are polynomials with nonzero constant terms, thus the entries of B are elements of F q [x]. Therefore, H is a GPM for some -MT code.
So far, Lemmas 1 and 2 show that H is a GPM of a -MT subcode of C ⊥ . Now we apply the standard dimension argument to show that this subcode is C ⊥ .

Lemma 3
The matrix H is a GPM of a -MT code of dimension n − k as an F qsubspace of F n q , where n = j=1 m j and k is the dimension of C.
What we proved in Lemmas 1, 2 and 3 can be summarized in the following theorem.
Theorem 6 Let C be a -MT code over F q with reduced GPM G and let A be the matrix that satisfies the identical equation of G. The polynomial matrix H given in Definition 3 is a GPM for C ⊥ .

Example 2
We continue with the (2, 1)-MT code C discussed in Example 1. From Theorem 5, C ⊥ is (2, 1)-MT over F 3 of length 60 and dimension 54. A GPM for C ⊥ can be obtained from Definition 3 and Theorem 6 as follows: The reduced GPM of C ⊥ is which can be obtained by reducing H to its Hermite normal form.
Since the class of MT codes contains QC, QT, and GQC codes as subclasses, the following special cases are direct consequences of Theorem 6.

Corollary 1 Let C be a QC code over F q of index , co-index m, and reduced GPM G = [g i, j ]. Let A denote the matrix satisfying the identical equation of G. Then C ⊥ is QC of index , co-index m, and a GPM
and d j = deg g j, j for 1 ≤ j ≤ .

Right, left, and two-sided Galois duals
In this section, we aim to generalize the result of Sect. 3 by replacing the Euclidean inner product with the Galois inner product. Furthermore, we present the two-sided Galois inner product of MT codes which has not been previously discussed in any study. Throughout this section, q = p e where p is a prime and e is a positive integer.
Recall that the Frobenius automorphism of F q , denoted σ , is defined by σ (α) = α p for each α ∈ F q . The Galois group of F q is finite, cyclic, and generated by σ . The least positive integer t such that σ t (α) = α for all α ∈ F q is t = e. Thus, the order of σ in the Galois group is e. Then σ extends to a ring automorphism of F q [x] by defining it on polynomials over F q as follows: σ a 0 + a 1 x + · · · + a m x m = σ (a 0 ) + σ (a 1 ) x + · · · + σ (a m ) x m .
In a natural way, σ extends to an automorphism of the ring of matrices over F q [x].
Let 0 ≤ μ < e and let υ = gcd (e, μ). Since υ divides e, F q has F p υ as a subfield. The automorphism σ μ of F q fixes an element α ∈ F q , i.e., σ μ (α) = α, if and only if α ∈ F p υ . In the ring of polynomials F q [x], σ μ only fixes all polynomials over F p υ . However, in the ring of matrices over F q [x], σ μ only fixes all matrices over F p υ [x]. Now, we examine the action of σ μ on vectors of F n q as an introduction to study its effect on linear codes over F q of length n.
Since σ μ (a + b) = σ μ (a) + σ μ (b) for any a, b ∈ F n q , σ μ defines an additive group automorphism on F n q . For a linear code C over F q of length n, let σ μ (C) = {σ μ (c) | c ∈ C}. The map σ μ : C → σ μ (C) is a group isomorphism; it is a group automorphism if and only if σ μ (C) = C, i.e., C is invariant under σ μ . We know that (a 1 , a 2 , . . . , a n ) ∈ F n q is fixed under σ μ if and only if a i ∈ F p υ for every 1 ≤ i ≤ n, where υ = gcd (e, μ). From the uniqueness of the reduced row echelon form of a generator matrix of a linear code, C is invariant under σ μ if and only if C has a basis that is a subset of F n p υ . Henceforth, for any 0 ≤ μ < e and some λ 1 , λ 2 , . . . , λ ∈ F q , let

. , m ). Let G be the reduced GPM of C and A the matrix that satisfies the identical equation of G. Then the reduced GPM of σ μ (C) is σ μ (G) and σ μ (A) is the matrix that satisfies the identical equation of σ μ (G).
Proof Since σ μ (C) is the image of C under a group isomorphism, σ μ (C) is an abelian group. Also, σ μ (C) is linear over F q because ασ μ (c) = σ μ σ e−μ (α) c ∈ σ μ (C) for every α ∈ F q and c ∈ C. In addition, σ μ (C) has dimension k because | C |=| σ μ (C) |.
Let C be -MT, then for any c ∈ C, we have
In [9], the Galois inner product is defined as an intrinsic generalization of the Euclidean inner product. Let a = (a 1 , a 2 , . . . , a n ) and b = (b 1 , b 2 , . . . , b n ) be two vectors in F n q . The Euclidean inner product of a and b, denoted a, b , is a symmetric bilinear form. For a fixed non-negative integer κ < e, define the κ-Galois inner product of a and b by the formula Clearly, a, b κ and b, a κ are not necessarily equal. For any α ∈ F q , αa, b κ = α a, b κ and a, αb κ = α p κ a, b κ . Thus, the Galois inner product is neither symmetric nor bilinear. In fact, a, b = a, b 0 for every a, b ∈ F n q . For a linear code C over F q of length n, the Euclidean inner product is used to define the Euclidean dual C ⊥ of C. On using the same imitation for the Galois inner product, we have to define two distinct duals: the right κ-Galois dual C ⊥ κ and the κ-left Galois dual C ⊥ κ .

Definition 5
Let κ < e be a non-negative integer and let C be a linear code over F q of length n. Define the right κ-Galois dual of C as follows: Theorem 8 Let κ < e be a non-negative integer and let C be a linear code over F q of length n and dimension k. The right κ-Galois dual C ⊥ κ of C is a linear code over F q of length n and dimension n − k. In fact, m 2 , . . . , m ) and has the reduced GPM σ e−κ (H), where H is the reduced GPM of C ⊥ .
Proof Observe that a, b κ = a, σ κ b for any a, b ∈ F n q . Thus C ⊥ κ = σ e−κ C ⊥ . Also observe that σ e−κ a, b = 0 if and only if a, σ κ Since σ e−κ : C ⊥ → C ⊥ κ is a group isomorphism, C ⊥ κ is linear of dimension n − k by Theorem 7. Now assume that C is -MT and recall from Theorem 5 that C ⊥ is -MT. Then by Theorem 7, C ⊥ κ is σ e−κ ( )-MT with reduced GPM σ e−κ (H).
Analogously to Definition 5 and Theorem 8, we define and investigate the left κ-Galois dual.

Definition 6
Let κ < e be a non-negative integer and let C be a linear code over F q of length n. The left κ-Galois dual C ⊥ κ of C is defined as the subset of F n q for which C ⊥ κ ⊥ κ = C, or in other words, Theorem 9 Let κ < e be a non-negative integer and let C be a linear code over F q of length n and dimension k. The left κ-Galois dual C ⊥ κ of C is a linear code over F q of length n and dimension n − k. In fact, m 2 , . . . , m ) and has the reduced GPM σ κ (H), where H is the reduced GPM of C ⊥ .
Proof Observe that a, b κ = a, σ κ b = σ κ σ e−κ a, b for any a, b ∈ F n q . Thus The last part can be proven in a way similar to that of Theorem 8.
In the literature and to the extent of our knowledge, left and right Galois duals were not jointly discussed in the same study. For instance, the Galois dual of a linear code is defined in [15] similar to our definition of the right Galois dual. Whereas in [5], the Galois dual is defined similar to our definition of the left Galois dual. We found it useful to include both Galois duals so that we can examine their interrelationships. One of these benefits is the following result.
Theorem 10 Let κ < e be a non-negative integer and let C be a linear code over F q . Then Proof By Theorems 8 and 9, we have

From 2 and 3, C
Throughout Theorems 11-16, we will adopt the following notations without introducing them. Let κ < e be a non-negative integer and choose a positive integer τ such that e | 4κτ . Let λ j ∈ F p υ for 1 ≤ j ≤ , where υ = gcd (e, 2κτ ). Let C be a -MT code over F q of block lengths (m 1 , m 2 , . . . , m ). The reduced GPM of C is G, while A is the matrix that satisfies the identical equation of G. The reduced GPM of C ⊥ is H, while B is the matrix that satisfies the identical equation of H. Our first goal is to provide some results for the codes C ⊥ κ and σ 2κτ C ⊥ κ .

Theorem 12
The reduced GPM of C ⊥ κ is σ e−κ (H), while the reduced GPM of Proof This is evident from Theorem 7 after noticing that C ⊥ κ = σ e−κ C ⊥ and There is another result to be obtained from Theorem 12 by applying σ e−κ and σ κ(2τ −1) to the identical equation of H. Namely, σ e−κ (B) and σ κ(2τ −1) (B) are the two matrices that satisfy the identical equations of σ e−κ (H) and σ κ(2τ −1) (H) respectively.

Theorem 13
The following conditions are equivalent.
By Theorem 7, σ 2κτ (G) is the reduced GPM of σ 2κτ (C). By the uniqueness of the reduced GPM, C = σ 2κτ (C) if and only if σ 2κτ (G) = G. Writing G = g i, j where g i, j ∈ F q [x] for 1 ≤ i, j ≤ , then σ 2κτ (G) = G if and only if σ 2κτ fixes g i, j for all i, j. That is, C = σ 2κτ (C) if and only if g i, j ∈ F p υ [x] for all i, j. lengths  (m 1 , m 2 , . . . , m ) and is invariant under σ 2κτ . Proof The first result is immediate from Theorem 11. Note that σ 4κτ acts as the identity on F n q because e | 4κτ . Hence, This inequality turns into equality because σ 2κτ is a group isomorphism. Hence, In this case, S has dimension deg (det (X)).
Proof Assume that S is invariant under σ 2κτ . Let P be the reduced GPM of S and let X be the matrix that satisfies the identical equation of P. By Theorems 2 and 12, there exists an upper triangular matrix Y such that P = Yσ e−κ (H) because S ⊆ C ⊥ κ . Since S is σ 2κτ -invariant, σ 2κτ fixes P. Thus P is over F p υ [x], then so is X because λ j ∈ F p υ for 1 ≤ j ≤ . But XYσ e−κ (H) = XP = σ e−κ (B) σ e−κ (H), then XY = σ e−κ (B). The dimension of S is immediate from Theorem 4. Conversely, suppose that S has a GPM Yσ e−κ (H) over F p υ [x]. We know that σ 2κτ fixes all matrices over F p υ [x]. Then Yσ e−κ (H) is a GPM for σ 2κτ (S). That is, σ 2κτ (S) = S.
In Theorem 16, we continue to use the general setting used above; however, the main result of this section will be an immediate consequence of it. Specifically, setting τ = 1 yields a result that fits the two-sided Galois dual of a MT code. This is described in Corollary 4.
Let C be a linear code. We define the two-sided Galois dual of C to be the intersection of its right and left Galois duals. Since C is linear, its two-sided Galois dual is linear because both C ⊥ κ and C ⊥ κ are linear. But if C is MT, then the two-sided Galois dual is not necessarily MT. Specifically, if C is -MT, then C ⊥ κ is σ e−κ ( )-MT and C ⊥ κ is σ κ ( )-MT. A sufficient condition to ensure that the two-sided Galois dual is MT is σ e−κ ( ) = σ κ ( ), or equivalently, λ j ∈ F p υ for 1 ≤ j ≤ , where υ = gcd (e, 2κ). Motivated by this condition, we start with the following definition for the two-sided Galois dual of a MT code.

Definition 7
Let e be a positive integer and let κ < e be a non-negative integer. Choose λ j ∈ F p υ for 1 ≤ j ≤ , where υ = gcd (e, 2κ). Let C be a -MT code over F q . Define the two-sided κ-Galois dual of C by C ⊥ κ ∩ C ⊥ κ .
The condition λ j ∈ F p υ for all 1 ≤ j ≤ in Definition 7 ensures that C ⊥ κ ∩ C ⊥ κ is σ κ -MT. This condition was previously used in Theorems 11-16 if τ = 1 is chosen. Furthermore, in case τ = 1, Theorem 10 shows that σ 2κτ C ⊥ κ can be replaced by C ⊥ κ . Therefore, Theorems 11-16 have proved the following result describing the two-sided Galois dual of a MT code.

Corollary 4
Let e be a positive integer and let κ < e be a non-negative integer such that e | 4κ. Define υ = gcd (e, 2κ) and let C be a (λ 1 , λ 2 , . . . , λ )-MT code over F q , where λ j ∈ F p υ for 1 ≤ j ≤ . Let G be the reduced GPM of C, let H be the reduced GPM of C ⊥ , and let B be the matrix that satisfies the identical equation of H. Then  On the other hand, the case of an invertible X is examined in Corollary 5.
(2) There is a pair (X, Y) that satisfies the conditions of Corollary 4. To see this, consider the set S = {(X i , Y i )} i∈I of all pairs that satisfy the first two conditions of Corollary 4. Clearly I , σ e−κ (B) ∈ S. Thus S is not empty. Moreover, S is totally ordered by deg (det (X i )). Since 0 ≤ deg (det (X i )) ≤ deg (det (B)) for every i ∈ I, there is a maximal element (X, Y) ∈ S. (3) It is not straightforward to determine matrices X and Y that satisfy the conditions given in Corollary 4. This is because these matrices have entries in different rings, F p υ [x] and F q [x]. We propose an auxiliary equation that may be useful in determining such matrices. If α ∈ F q , the trace of α, written Tr (α), is defined by For any a, b ∈ F p υ and α, β ∈ F q , we have Tr (aα + bβ) = aTr(α) + bTr(β). We can extend Tr to an additive group homomorphism from F q [x] to F p υ [x] by defining Tr a 0 + a 1 x + · · · + a m x m = Tr(a 0 ) + Tr(a 1 )x + · · · + Tr(a m )x m .
Similarly, for matrices over F q [x], Tr defines a group homomorphism.
. Suppose X and Y as defined in Corollary 4. For any 0 ≤ i ≤ (e − υ)/υ, the automorphism σ iυ fixes X in the ring of matrices over F q [x]. Therefore, By summing (11) over all values of i, we get the auxiliary equation All matrices in (12) are over F p υ [x]. In Example 3 below, we will indicate how to use (12) to determine matrices that satisfy the conditions given in Corollary 4. (4) Some extra conditions must be taken into account when one aims to make Yσ e−κ (H) the reduced GPM of C ⊥ κ ∩ C ⊥ κ . In this situation, X = x i, j is the matrix that satisfies the identical equation of the reduced GPM. Then for each 1 ≤ i ≤ , x i,i is a nonzero monic polynomial and deg It is worthwhile to present a complete example illustrating the process of determining the reduced GPM of the right Galois dual, the left Galois dual, and the two-sided Galois dual of a MT code.
More precisely, C ⊥ 3 ∩ C ⊥ 3 = C 1 ⊕ 0 ⊕ 0, where C 1 is the cyclic code of length 3 over F 16 with generator polynomial θ 10 + x and 0 is the zero code of length 4.
We conclude this section with an application of Corollary 4 concerning when the right and the left Galois duals trivially intersect. Obviously, C ⊥ κ ∩ C ⊥ κ = {0} if and only if the first two conditions of Corollary 4 can only be satisfied by an invertible X, hence zero is the maximum of deg (det (X)). We remark that these two conditions are always satisfied by an invertible X, for instance X = I and Y = σ e−κ (B). However, if these two conditions are never satisfied except for an invertible X, then C ⊥ κ ∩ C ⊥ κ = {0}. The following is a particular case that requires the code dimension to be half the code length.

Corollary 5
Let e be a positive integer and let κ < e be a non-negative integer such that e | 4κ. Define υ = gcd (e, 2κ) and let C be a (λ 1 , λ 2 , . . . , λ )-MT code over F q of length n and dimension n/2, where λ j ∈ F p υ for 1 ≤ j ≤ . Let H be the reduced GPM of C ⊥ and let B be the matrix that satisfies the identical equation of H. Let X and Y be upper triangular matrices over F p υ [x] and F q [x], respectively, such that 1. Yσ e−κ (H) is a matrix over F p υ [x], and 2. XY = σ e−κ (B).
Then F n q = C ⊥ κ ⊕ C ⊥ κ if and only if the above conditions can only be satisfied by an invertible X.
Proof From Corollary 4, there exist X and Y such that deg (det (X)) is the dimension of C ⊥ κ ∩ C ⊥ κ . If the given conditions can only be satisfied by an invertible X, then C ⊥ κ ∩ C ⊥ κ has dimension zero. Therefore, C ⊥ κ ∩ C ⊥ κ = {0} and F n q = C ⊥ κ ⊕ C ⊥ κ because both C ⊥ κ and C ⊥ κ have dimension n/2.
Conversely, suppose that X and Y satisfy the given conditions. From Theorem 15, there exists a subcode S of C ⊥ κ ∩C ⊥ κ such that S has dimension deg (det (X)). Assume that F n q = C ⊥ κ ⊕ C ⊥ κ . Then S ⊆ C ⊥ κ ∩ C ⊥ κ = {0}, and hence deg (det (X)) = 0. That is, X is invertible.
We found that C has parameters [12, 6, 2] over F 81 . The matrix that satisfies the identical equation of G is from which we conclude that C has a dimension equal to half the code length. By Theorems 5 and 6, the Euclidean dual C ⊥ of C is θ 30 , θ 60 -MT with a GPM whose reduced form is H = θ 15 + x 2 θ 75 + x 2 0 θ 50 + θ 5 x 2 + x 4 , and the matrix that satisfies the identical equation of H is B = θ 55 + x 2 2 0 θ 50 + θ 45 x 2 + x 4 .
From (15), θ 45 + x 2 divides y 11 and (θ 5 + x 2 )(θ 65 + x 2 ) divides y 11 because it is required to make y 11 θ 5 + x 2 and y 22 θ 25 + x 2 θ 45 + x 2 elements in F 9 [x]. It follows from (16) that x 11 = x 22 = 1 and, hence, X is invertible. By Corollary 5, F 12 81 can be written as the direct sum of the right and left 1-Galois duals of C, that is, Funding Open access funding provided by The Science, Technology & Innovation Funding Authority (STDF) in cooperation with The Egyptian Knowledge Bank (EKB).
Declarations The author has no conflicts of interest to declare. The author has no relevant financial or non-financial interests to disclose.
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