A construction of self-dual skew cyclic and negacyclic codes of length $n$ over $\FF_{p^n}$

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A construction of self-dual skew cyclic and negacyclic codes of length n over _p

Aicha Batoul, Delphine Boucher, Ranya Boulanouar

To cite this version:

Aicha Batoul, Delphine Boucher, Ranya Boulanouar. A construction of self-dual skew cyclic and negacyclic codes of lengthnover _pⁿ. WAIFI 2020: Arithmetic of Finite Fields, Jul 2020, RENNES, France. �hal-02904416�

A construction of self-dual skew cyclic and negacyclic codes of length n over F

.

Aicha Batoul ^∗ Delphine Boucher^† Ranya Djihad Boulanouar ^‡

July 19, 2020

Abstract

The aim of this note is to give a construction and an enumeration of self-dualθ-cyclic and θ-negacyclic codes of length n overF^pn where pis a prime number and θ is the Frobenius automorphism overFpⁿ. We use the notion of isodual codes to achieve this construction.

1 Introduction

Isodual codes ( [17]) have been recently studied on many aspects ( [2], [3], [1]). Meanwhile, in [5], a construction and an enumeration formula for self-dualθ-cyclic andθ-negacyclic codes of even length nover Fp² were given in the case when pis a prime number and θ is the Frobenius automorphism over Fp². The aim of this note is to give a construction and an enumeration formula for self-dualθ-cyclic andθ-negacyclic codes of lengthnoverFpⁿ whenθis the Frobenius automorphism overFpⁿ. To this end, we will use and develop the notion of (θ, ν)-isodual codes which form a subfamily of the family of isodual codes. Lastly we will consider the construction of some self-dual Gabidulin evaluation codes.

The text is organized as follows. In Section 2 we define the notion of (θ, ν)-isodual codes over Fq where θ is an automorphism of Fq and ν belongs to F^∗q. We recall the definitions of (θ, a)-constacyclic,θ-cyclic and θ-negacyclic codes and some generalities on the dual of a (θ, a)- constacyclic code. Then we characterize (θ, ν)-isodual θ-cyclic and θ-negacyclic codes thanks to an equation satisfied by the skew check polynomials of the codes. In Section 3 we consider the special case whenqis equal topⁿwherepis a prime number andθis the Frobenius automorphism overFpⁿ. After having given a necessary and sufficient condition for the existence of (θ, ν)-isodual θ-cyclic and θ-negacyclic codes, we give a construction and an enumeration formula for (θ, ν)- isodual and self-dual θ-cyclic and θ-negacyclic codes. In Section 4, we consider a subclass of self-dualθ-cyclic codes overF^pn which are self-dual Gabidulin codes. We parametrize this family by a parameter which satisfies a polynomial system.

2 Some generalities on isodual skew codes

We first recall that alinear codeC of lengthnand dimensionkoverFq is a subspace of dimension kofFⁿq. Agenerator matrixGofC is ak×nmatrix with coefficients inF^q and rankksuch that

∗Faculty of Mathematics, University of Science and Technology Houari Boumedienne (USTHB), 16111 Bab Ezzouar, Algiers, Algeria

†Univ Rennes, CNRS, IRMAR - UMR 6625, F-35000 Rennes, France

‡Faculty of Mathematics, University of Science and Technology Houari Boumedienne (USTHB), 16111 Bab Ezzouar, Algiers, Algeria

C={m G|m∈F^kq}. Furthermore the dual ofC isC^⊥ ={x∈Fⁿq | ∀c∈C, < x, c >= 0}where for x = (x₀, . . . , x_n−1), y = (y₀, . . . , y_n−1) in Fⁿq, < x, y >:= Pn−1

i=0 x_iy_i is the Euclidean scalar product ofxandy. Isodual codes ( [17]) have been recently studied on many aspects ( [2], [3], [1]).

Definition 1 ( [17] page 199). A codeC with generator matrixGisisodual if it is equivalent to its dual. That means that there exists a monomial matrixD such thatG·D is a generator matrix of the dualC^⊥ of C.

In what follows, we define a special class of isodual codes which are parameterized by an automorphismθofFq and an elementν ofF^∗q.

Definition 2. Consider n ∈ N^∗, ν ∈ F^∗q and θ ∈ Aut(Fq). A linear code C of length n and generator matrix Gis a (θ, ν)-isodual code ifG·D is a generator matrix of C^⊥ where D is the n×n diagonal matrix with diagonal coefficientsν, θ(ν), . . . , θⁿ⁻¹(ν).

Remark 1. A codeCis self-dual if and only if there existsνfixed byθsuch thatCis(θ, ν)-isodual.

Recall that ifθis an automorphism ofFq, the skew polynomial ringRis defined asR=Fq[X;θ]

under usual addition of polynomials and where multiplication is defined by the commutation law :

∀a∈Fq, X·a=θ(a)X( [16]). The ringRis noncommutative unlessθis the identity automorphism on Fq. The ring R is right-Euclidean and left-Euclidean. For f = P

aiXⁱ in R and α in Fq, the evaluation f(α) of f at α is the remainder in the right division off by X −α. We have f(α) =P

iaiNi(α) whereNi(x) :=xθ(x)· · ·θⁱ⁻¹(x) (see [14]). Recall also that ifq=pⁿ andθis the Frobenius automorphism, then the center ofRisFp[Xⁿ] .

ForainF^∗q andθinAut(Fq), a (θ, a)-constacycliccodeCof lengthnand dimensionkis a left R-submoduleRg/R(Xⁿ−a)⊂R/R(Xⁿ−a) wheregis a monic skew polynomial of degreen−k right-dividingXⁿ−ainR( [7]). That means that a wordc= (c₀, . . . , c_n−1)∈Fⁿq belongs toC if and only if the skew polynomial gright-divides the skew polynomial c₀+c₁X+· · ·+c_n−1Xⁿ⁻¹ in R. The skew polynomial g is called the skew generator polynomial of C. The monic skew polynomialhdefined by

Θⁿ(h)·g=Xⁿ−a (1)

is calledskew check polynomialofC.

The (θ, a)-constacyclic code C is denoted C = (g)^a_n,θ. If a = 1, the code is θ-cyclic and if a=−1, the code isθ-negacyclic.

A generator matrix of Cis

G=













g0 g1 . . . . . . 1 0 . . . 0 0 θ(g₀) θ(g₁) . . . . . . 1 . .. ...

... . .. . .. . .. 0

0 . . . 0 θ^k−1(g₀) θ^k−1(g₁) . . . . . . 1













. (2)

The skew reciprocal polynomialof h= Σ^k_i=0h_iXⁱ ∈R of degreek ish^∗ = Σ^k_i=0θⁱ(h_k−i)Xⁱ. If h₀ 6= 0, theleft monic skew reciprocal polynomial of his h^\= _θk(h¹

0)h^∗. The following technical lemma will be useful later. We will use the application Θ : R 7→ R given by Pk

i=0aiXⁱ 7→

Pk

i=0θ(a_i)Xⁱ.

Lemma 1 (Lemma 1 of [8]). Considerθ∈Aut(F^q),R=F^q[X;θ],handg inR. Then(h·g)^∗= Θ^deg(h)(g^∗)·h^∗.

Example 1. Considern= 4,F2⁴ =F2(a)witha⁴+a+ 1 = 0andR=F2⁴[X;θ]. We have X⁴+ 1 = (X²+a⁵X+a⁵)·(X²+a⁵X+a¹⁰)

therefore the skew polynomialg₁ =X²+a⁵X+a¹⁰ generates a θ-cyclic codeC₁ of length 4 and dimension 2 over F2⁴. As Θ⁴ is the identity overF2⁴, the skew check polynomial of the code is h₁=X²+a⁵X+a⁵.

We have

X⁴+ 1 = (X²+aX+a¹⁴)·(X²+a⁴X+a)

therefore the skew polynomial g2 =X²+a⁴X+a generates a θ-cyclic code C2 of length 4 and dimension 2 overF2⁴ with skew check polynomialh2=X²+aX+a¹⁴.

The following proposition describes the dual of a (θ, a)-constacyclic code.

Proposition 1(Theorem 1 and Lemma 2 of [8], Proposition 1 of [6]). Consider n∈N^∗,a∈F^∗q, θ∈Aut(Fq)andCa(θ, a)-constacyclic code of lengthnwith skew generator polynomialgand skew check polynomial h. Then the dual C^⊥ of C is a (θ,1/a)-constacyclic code with skew generator polynomialh^\.

Proof. (proof of Proposition 1 of [6]) We consider the equality (1) inR=F^q[X;θ] and we multiply both members of this equality byhon the right. We get Θⁿ(h)·g·h= (Xⁿ−a)·hand we deduce from this equality that Θⁿ(h)·(Xⁿ−g·h) = a·h. As the skew polynomials Θⁿ(h) and a·h have the same degrees, the skew polynomial Xⁿ−g·h is a constant that we will denoteλand Θⁿ(h)·λ−a·h= 0. As the leading coefficient of Θⁿ(h)·λ−a·his equal to θ^k(λ)−a, we get thatλ=θ^−k(a).

Furthermore, as Θⁿ(h)·g=Xⁿ−a, according to Lemma 1, we have−_a¹Θ^k−n(g^∗)·h^∗=Xⁿ−¹_a. Thereforeh^\ right-divides Xⁿ−¹_a and is the skew generator polynomial of a (θ,¹_a)-constacyclic code of lengthn.

A quick computation gives that for all (i, j) in{0, . . . , k−1} × {0, . . . , n−k−1}, the Euclidean scalar product of the words associated toXⁱ·g andX^j·h^∗ is equal toθⁱ((g·h)_j−i+k). Therefore the scalar product is equal to 0 and the words of the code (g)^a_n,θ are orthogonal to the words of the code (h^\)^1/a_n,θ.

In what follows, we characterize (θ, a)-constacyclic codes which are (θ, ν)-isodual.

Proposition 2. Consider k ∈N^∗, n= 2k, ν ∈ F^∗q, a∈Fq, θ ∈Aut(Fq),R =Fq[X;θ], h∈R monic. The (θ, a)-constacyclic code of length n, dimension k and skew check polynomial h is (θ, ν)-isodual if and only if

Θⁿ(h)·θ^k(ν)·h^\·1

ν =Xⁿ−a. (3)

In this case we have a²=θⁿ(ν)/ν.

Proof. ConsiderC= (g)^a_n,θthe (θ, a)-constacyclic code of lengthn= 2k, dimensionk, skew check polynomial hand skew generator polynomial g. According to (1), we have Θⁿ(h)·g =Xⁿ−a.

Therefore, the relation (3) is satisfied if and only ifh^\= ˜g where ˜g=θ^k(1/ν)·g·ν.

Let us prove that C is (θ, ν)-isodual if and only if h^\ = ˜g. As g right-divides Xⁿ −a, ˜g right-divides Xⁿ −a˜ where ˜a = a_θn^ν(ν). Therefore we can consider the (θ,a)-constacyclic code˜ of length nand skew generator polynomial ˜g. Furthermore, according to Proposition 1,C^⊥ is a (θ,1/a)-constacyclic code of lengthnwith skew generator polynomialh^\.

Let us prove thatC is (θ, ν)-isodual if and only if (h^\)^1/a_n,θ = (˜g)^˜a_n,θ. Denote g=Pn−k

i=0 g_iXⁱ=Pk

i=0g_iXⁱ and ˜g=Pk

i=0g˜_iXⁱ. We have ˜g_i=θ^k(1/ν)g_iθⁱ(ν) for all iin {0, . . . , k}. Therefore a generator matrix of (˜g)^˜a_n,θ is

G˜=













˜

g₀ ˜g₁ . . . . . . 1 0 . . . 0 0 θ(˜g₀) θ(˜g₁) . . . . . . 1 . .. ...

... . .. . .. . .. 0

0 . . . 0 θ^k−1(˜g0) θ^k−1(˜g1) . . . . . . 1













=G·D

whereGis given by (2) andDis the diagonal matrix with diagonal elementsν, θ(ν), . . . , θⁿ⁻¹(ν).

According to Definition 2, the code C is (θ, ν)-isodual if and only if a generator matrix of C^⊥ = (h^\)^1/a_n,θ is G·D. As G·D = ˜G is a generator matrix of (˜g)^a_n,θ^˜ , we obtain that C is (θ, ν)-isodual if and only if (h^\)^1/a_n,θ = (˜g)^˜a_n,θ.

Lastly as ˜gright-dividesXⁿ−a_θn^ν(ν) andh^\right-dividesXⁿ−_a¹, we obtaina²=θⁿ(ν)/ν.

In the case whenθ is the Frobenius automorphism overFq andq=pⁿ wherepis prime andn is the length of the code we obtain the following corollary that will be useful in next section.

Corollary 1. Consider k ∈N^∗, n = 2k, p a prime number, ν ∈ F^∗pⁿ, a ∈ Fpⁿ, θ :x 7→x^p ∈ Aut(Fpⁿ), R=Fpⁿ[X;θ],h∈R monic. The (θ, a)-constacyclic code of length n and skew check polynomialhis(θ, ν)-isodual if and only if

h·θ^k(ν)·h^\· 1

ν =h^\·1

ν ·h·θ^k(ν) =Xⁿ−a. (4) Furthermore,a²= 1.

Proof. As θ is the Frobenius automorphism over Fpⁿ, the order of θ is equal to n. Therefore Θⁿ(h) = hand θⁿ(ν) =ν. According to Proposition 2, the (θ, a)-constacyclic code of length n and skew check polynomialhis (θ, ν)-isodual if and only ifh·θ^k(ν)·h^\·¹_ν =Xⁿ−a. In this case a²= 1. ThereforeXⁿ−ais central inR, and we haveh·θ^k(ν)·h^\·_ν¹ =Xⁿ−a=h^\·_ν¹·h·θ^k(ν).

Example 2. (Example 1 continued) The left monic skew reciprocal polynomial of h1 = X²+ a⁵X +a⁵ is h^\₁ = X² +a⁵X +a¹⁰ and X⁴+ 1 = (X²+a⁵X +a⁵)·(X²+a⁵X +a¹⁰) = (X²+a⁵X+a¹⁰)·(X²+a⁵X+a⁵), therefore theθ-cyclic codeC₁ with skew check polynomialh₁ is self-dual.

The left monic skew reciprocal polynomial of h₂ = X²+aX +a¹⁴ is h^\₂ =X²+a⁶X +a⁴. Furthermore,X⁴+1 = (X²+aX+a¹⁴)·(X²+a⁴X+a) = (X²+aX+a¹⁴)·_a¹₄·(X²+a⁶X+a⁴)·_a¹₁₄, therefore the θ-cyclic code C2 with skew check polynomialh2 is(θ, a¹⁴)-isodual.

Lastly, we consider below a technical lemma which will be useful later and which deals with the factorization of skew polynomials right-dividing Xⁿ±1 in Fpⁿ[X;θ] where θ is the Frobe- nius automorphism. These skew polynomials belong to a wide class of skew polynomials, called Wedderburn polynomials, which have been extensively studied (see Theorem 6.4 of [11] for the factorizations of these skew polynomials). Lemma 2 can be directly deduced from Theorem 6.4 of [11] as well as from Proposition 2.2.2. of [10]. We propose here a proof very specific to our special case.

Lemma 2. Consider n∈ N^∗, pa prime number, θ :x7→x^p ∈Aut(Fpⁿ), R =Fpⁿ[X;θ], f in R of degree dand∈ {−1,1} such that f right-dividesXⁿ−in R. Then f is the product of d linear factors right-dividingXⁿ−and

#{(α₁, . . . , α_d)∈F^dpⁿ |f = (X+α₁)· · ·(X+α_d)}=

d

Y

i=1

pⁱ−1 p−1.

Proof. Considery₁, . . . , y_n inFpⁿlinearly independent overFp. ConsiderξinFpⁿsuch thatX−ξ right-divides Xⁿ−, which meansN_n(ξ) = . Denoteα₁ :=ξ^θ(y_y¹⁾

1 , . . . , α_n =ξ^θ(y_yⁿ⁾

n . According to [14], the least common left multiple of X−α₁, . . . , X−α_n is lclm_1≤i≤n(X −α_i) = Xⁿ−. Asf right-dividesXⁿ−, according to Theorem 4 of [16], there existβ₁, . . . , β_d inFpⁿ such that f = lclm_1≤i≤d(X−βi). FurthermoreNn(βi) = =Nn(ξ). Therefore according to Theorem 27 of [13], there exists zi in F^∗pⁿ such that βi = ξ^θ(z_zⁱ⁾

i . According to [14], z1, . . . , zd are linearly independent over F^p. Denote f = Pd

i=0aiXⁱ, we have {α ∈ F^pn | f(α) = 0} = {α =ξ^θ(y)_y = ξy^p−1 ∈ Fpⁿ | L(y) = 0} where L(y) := Pd

i=0aiNi(ξ)θⁱ(y). As the equation L(y) = 0 has d solutions (z1, . . . , zd) in Fpⁿ linearly independent overFp, there arep^d−1 nonzero y inFpⁿ such that L(y) = 0. Therefore {α∈Fpⁿ | X−αright-divides f} has (p^d−1)/(p−1) elements. We conclude using an inductive argument.

Remark 2. It could be noted that the number

d

Y

i=1

pⁱ−1

p−1 is the size of the general linear group GL(d, p) modulo diagonal matrices, and corresponds to choosing a set of 1-dimensional vector spaces spanning a d-dimensional vector space.

Example 3. (Example 1 continued) The skew polynomialh1=X²+a⁵X+a⁵has3factorizations into the product of linear monic skew polynomials, namelyh1= (X+a¹⁴)·(X+a⁶) = (X+a¹¹)·(X+ a⁹) = (X+1)·(X+a⁵). The skew polynomialh2=X²+aX+a¹⁴has also3factorizations into the product of linear monic skew polynomials, namelyh2= (X+a⁷)·(X+a⁷) = (X+a¹¹)·(X+a³) = (X+a¹⁰)·(X+a⁴).

3 Construction and enumeration of (θ, ν)-isodual θ-cyclic and θ-negacyclic codes of length n over F

The aim of this section is to construct and to enumerate self-dualθ-cyclic andθ-negacyclic codes of length nover Fpⁿ where θ is the Frobenius automorphism. Note that in this setting,θ-cyclic codes are calledGabidulinp-cyclical codes(page 6 of [12]).

To achieve this construction, we will consider θ-cyclic andθ-negacyclic codes which are (θ, ν)- isodual.

We introduce some notation. Consider R = F^pn[X;θ]. We will denote, for ∈ {−1,1} and ν∈F^∗q :

H_ν,:={h∈R|hmonic, h^\· 1

ν ·h·θ^k(ν) =Xⁿ−}.

According to Corollary 1, the setHν,is the set of the skew check polynomials of (θ, ν)-isodual (θ, )-constacyclic codes. Following Remark 1, the setH1,is the set of the skew check polynomials of self-dual (θ, )-constacyclic codes.

3.1 Necessary and sufficient existence condition for(θ, ν)-isodualθ-cyclic and θ-negacyclic codes of length n over Fpⁿ

In [4] a necessary and sufficient condition for the existence of self-dual (θ, )-constacyclic codes over a finite fieldFq was derived where θ is an automorphism ofFq and ∈ {−1,1}. In what follows we give a necessary and sufficient condition for the existence of (θ, ν)-isodual (θ, )-constacyclic codes of lengthnoverFpⁿ where pis a prime number andθ is the Frobenius automorphism.

Proposition 3. Considerk∈N^∗,n= 2k,pa prime number,θ:x7→x^p∈Aut(Fpⁿ),∈ {−1,1}.

(i) Ifp= 2, then there exists a(θ, ν)-isodual θ-cyclic code of lengthn for allν inF^∗pⁿ.

(ii) Ifpis odd, then forν ∈F^∗pⁿ, there exists a (θ, ν)-isodual(θ, )-constacyclic code of length 2k if and only if

ν^pn−12 =−(−1)^kp−12 .

Proof. According to Corollary 1, there exists a (θ, ν)-isodual (θ, )-constacyclic code of lengthnif and only if the setHν, is nonempty.

• Assume that p = 2 (therefore = 1) and ν ∈ F^∗2ⁿ. Consider α in F2ⁿ such that α² = 1/ν^2k−1= _θk^ν(ν) andh=X^k+α. We have

h^\·¹_ν ·h·θ^k(ν) =

X^k+ 1 θ^k(α)

·

X^k+θ^k(ν)α ν

= X^2k+θ^k 1

α+αθ^k(ν) ν

X^k+θ^k(ν)α θ^k(α)ν.

Asν/θ^k(ν) =α², we obtain h^\· 1

ν ·h·θ^k(ν) =X^2k+ 1 αθ^k(α).

As θ(α) = α² = _θk^ν_(ν), we obtain θ^k+1(α) = ^θk_ν^(ν), θ(α)θ^k+1(α) = 1 and αθ^k(α) = 1.

Therefore the skew polynomial hbelongs toHν,.

• Assume thatpis odd andν^pn−12 =−(−1)^kp−12 . We have

(−^θk_ν^(ν))^(pn−1)/2 = (−1)^{(pn−1)/2θk}_ν^(ν(pn−1)/2^(pn−1)/2)

= (−1)^(pn−1)/2

= 1 (becausepis odd, thereforep²≡1 (mod 4) andpⁿ ≡1 (mod 4)).

Therefore−_θk^ν_(ν) is a square. Considerαsuch that −_θk^ν_(ν) =α². Consider h=X^k+αin R.

h^\·¹_ν ·h·θ^k(ν) =

X^k+ 1 θ^k(α)

·

X^k+θ^k(ν)α ν

= X^2k+θ^k 1

α+αθ^k(ν) ν

X^k+θ^k(ν)α θ^k(α)ν. Asν/θ^k(ν) =−α², we obtain

h^\· 1

ν ·h·θ^k(ν) =X^2k+θ^k(ν)α θ^k(α)ν. Furthermore

θ^k(ν)α

θ^k(α)ν = ^θk_ν^(ν)

−^θk_ν^(ν)pk−1₂

(because−θ^k(ν)/ν= 1/α²)

= (−1)^pk−12 ν^pn−12

= − (becauseν^pn−12 =−(−1)^kp−12 ).

Therefore

h^\·¹_ν ·h·θ^k(ν) = Xⁿ−.

• Assume that p is odd and that there exists a (θ, )-constacyclic code of length 2k which is (θ, ν)-isodual. Consider h its skew check polynomial and h0 the constant term of h.

Necessarily the degree ofhis equal tok.

As the code is (θ, )-constacyclic of length 2k,hright-dividesX^2k−. As the code is defined overFp^2k withθ:x7→x^p,X^2k−is central of degree 1 inF^p[X^2k] andhis the product of k linear skew polynomialsX+α1, . . . , X+αk right-dividing X^2k−. Thereforeh0 is the product of thekconstant termsα1, . . . , αk. AsN2k(−αi) =, we have :

N2k(h0) =^k=Nk(h0)Nk(θ^k(h0)).

As the code is (θ, ν)-isodual, we haveh^\·_ν¹·h·θ^k(ν) =Xⁿ− and h0θ^k(ν) +θ^k(h0)ν = 0.

AsNk(h0)Nk(θ^k(h0)) =^k, we obtainNk(h0)²(−)^{k N}_N^2k(ν)

k(ν)² =^k and