Hulls of cyclic serial codes over a finite chain ring

Finite Fields and Their Applications 77 (2022) 101950 Contents lists available at ScienceDirect Finite Fields and Their Applications www.elsevier.com/locate/ffa Hulls of cyclic serial codes over afinite chain ring Sarra Talbi a, Aicha Batoul a, Alexandre Fotue Tabue b, Edgar Martínez-Moro c,∗,1 aFaculty of Mathematics USTHB, University of Science and Technology of Algiers, Algeria bDepartment of Mathematics, HTTC Bertoua, The University of Ngaoundéré, Cameroon cInstitute of Mathematics, University of Valladolid, Castilla, Spain a r t i c l e i n f o a b s t r a c t Article history: Received 13 February 2021 Received in revised form 21 September 2021 Accepted 14 October 2021 Available online xxxx Communicated by W. Cary Huffman MSC: 13B02 94B05 Keywords: Finite chain ring Cyclotomic coset Cyclic serial code Galois dual code Average pr-dimension In this paper, we explore some properties of hulls of cyclic serial codes over a finite chain ring and we provide an algorithm for computing all the possible parameters of the Euclidean hulls of that codes. We also establish the average pr-dimension of the Euclidean hull, where Fpris the residue field of R, as well as we give some results of its relative growth. © 2021 The Author(s). Published by Elsevier Inc. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/). *Corresponding author. E-mail addresses: [email protected] (S. Talbi), [email protected] (A. Batoul), [email protected] (A. Fotue Tabue), [email protected] (E. Martínez-Moro). 1This author is partially funded by the Spanish Research Agency (AEI) under Grant PGC2018-096446B-C21. https://doi.org/10.1016/j.ffa.2021.101950 1071-5797/© 2021 The Author(s). Published by Elsevier Inc. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/). 2S. Talbi et al. / Finite Fields and Their Applications 77 (2022) 101950 1. Introduction The Euclidean hull is defined to be the intersection of a code and its Euclidean dual. It was originally introduced in [1]to classify finite projective planes. Knowing the hull of a linear code is also a key point to determine the complexity of some algorithms for investigating permutations of two linear codes and computing the automorphism group of the code [11,15,16]. In general, those algorithms have been proved to be very effective if the size of the Euclidean hull is small. In the case of codes over finite fields, Sendrier [17] established the number of linear codes of length nwith a fix dimension Euclidean hull, also Skersys [19] discussed the average dimension of the Euclidean hull of cyclic codes. Later, Sangwisut et al. [18] determined the dimension of the Euclidean hull of cyclic and negacyclic codes of length nover a finite field. Furthermore, in [9]the authors gave the average Euclidean hull dimension of negacyclic codes over a finite field. Recently, the concept of the Euclidean hulls has been generalized to cyclic codes of odd length over Z4 in [10] where the authors provided an algorithm to determine the type of the Euclidean hull of cyclic codes over Z4. An important class of linear codes over rings is the class of cyclic codes and they have been extensively studied, see for example [4,5,7,13,14]. In particular, Dinh and Permouth [4]gave the algebraic structure of simple root cyclic codes over finite chain rings Rand in [13]this was generalized to multivariable cyclic codes. Free cyclic serial codes have been determined by using cyclotomic cosets and trace map over finite chain rings [5]. It is clear that the Euclidean hull of cyclic codes is also cyclic, two special families of cyclic codes are of great interest, namely linear complementary dual codes, which are codes whose Euclidean hull is trivial (see for example [3]) and self-orthogonal codes, which are linear codes whose Euclidean hulls are the whole code (see for example [20,2]). These works motivate us to study the hulls of cyclic codes over finite chain rings. In this paper, we focus on the study of the hulls of cyclic codes of length nover a finite chain ring Rof parameters (p, r, a, e, r)such that nand pare coprime. This is the serial case stated in [13], i.e. the cyclic codes over Rwhose length nis coprime with pare serial modules over R. We will generalize the techniques used in [10](for Z4)to obtain the parameters and the average pr-dimensions Euclidean hull of cyclic serial codes over finite chain rings. The paper is organized as follows. In Section 2, some preliminary concepts and some basic results are recalled. In Section 3, we characterize Galois hulls of cyclic serial code over finite chain rings. Section 4shows the parameters and the q-dimensions of the Euclidean hull of cyclic serial codes. Finally, the average dimension of the Euclidean hull of cyclic serial codes is computed in Section 5. 2. Preliminaries 2.1. Chain rings For an account on the results on finite rings in this section check [12]. Throughout this paper, pis a prime number, a, e, r, sare positive integers and Zpais the residue ring S. Talbi et al. / Finite Fields and Their Applications 77 (2022) 101950 3 of integers modulo pa. Rwill denote a finite commutative chain ring of characteristic pa, of nilpotency index s, and of residue field Fq(where q=pr). We will denote its maximal ideal by J(R)and R×will denote its multiplicative group. Note that since R is a chain ring it is a principal ideal ring, thus we will denote by θa generator of J(R), the ideals of Rform a chain under inclusion {0} =J(R)sJ(R)s−1···J(R) R and J(R) =θtRfor 0 ≤t <s. The ring epimorphism π:R→R/J(R) ≃Fqnaturally extends a ring epimorphism from R[X]to Fpr[X]and on the other hand it naturally induces an R-module epimorphism from Rnto (Fpr)n. As an abuse of notation we will denote both mappings by π. A monic polynomial fis basic-irreducible over Rif π(f)is irreducible over Fpr. We will denote by GR(pa, r)the Galois ring of characteristic paand cardinality pra. It is well known that, for a given finite chain ring Rthere is a 5-tuple (p, a, r, e, s)of positive integers, the so-called parameters of R, such that R=GR(pa, r)[θ], and θ =J(R), θe∈ p(Zpa[θ])×and θs−1=θs=0 R. From now on, we will denote as Sdthe subring of Rsuch that Sd:= GR(pa, d)[θ]and dis a divisor of r. The Teichmüller set of Rwill be denoted as Γ(R)and it is defined as Γ(R) ={0} ∪{a ∈R:apr−2=apr−1=1}. It is the only cyclic subgroup of R×isomorphic to the multiplicative group of Fpr. For each element a in R, there is a unique (a0, a1, ··· , as−1)in Γ(R)ssuch that a =a0+a1θ+···+as−1θs−1. Let Rand Sbe two finite commutative chain rings, we say that we say that Ris an extension of Sand we denote it by S|Rif S⊂Rand 1R=1 S. We say that the extension is separable if J(S)R=J(R). The Galois group of the extension S|R, denoted AutS(R), is the group of all the automorphisms γof Rwhose restriction γ|Sof γto S, is the identity map of R. A separable extension is called Galois if {r∈R:(∀γ∈AutS(R))(γ(r) = r)} =S. This condition is equivalent to the condition Ris ring-isomorphic to S[X]/f, where fis a monic basic irreducible polynomial in S[X], see [22, Section 4][12, Theorem XIV.8]. Let dbe positive divisor of r, and let us consider S=Zpa[θ], R=GR(pa, r)[θ], Sd=GR(pa, d)[θ], and GSub(S|R):={Sd:dis a divisor of rand Zpa[θ]⊆Sd}. It is well known that AutS(R)is a cyclic group generated by the Frobenius automorphism σ:R→Rgiven by: σs−1  t=0 atθt= s−1  t=0 ap tθt, and therefore, the set Sub(AutS(R)) of subgroups of AutS(R)is given by Sub(AutS(R)) = {σd:dis a divisor of r}. In [6], the authors established the Galois correspondence (Stab; Fix) between GSub(S|R) and Sub(AutS(R)) as follows Stab :GSub(S|R) →Sub(AutS(R)) and Fix :Sub(AutS(R)) →GSub(S|R) where Stab(Sd) =σdand Fix(σd) =Sd, where dis a divisor of r(recall that q=pr). 4S. Talbi et al. / Finite Fields and Their Applications 77 (2022) 101950 Given a divisor dof r, from [12, Theorem XV.2], σdis the only automorphism in AutS(R)such that σd◦π=π◦σd, where σis a generator of AutFp(Fpr). The trace map Td:S→Sdof the ring extension R|Sdis defined by Td:= r d−1  i=0 σid, and the trace map Td:Fpr→Fpdof the field extension Fpr|Fpdis defined by Td:= r d−1  i=0 σid. It is well known that Td:R→Sqis an epimorphism of Sd-modules and Td:Fpd→Fpris an epimorphism of vector spaces over Fpd. Hence, for any divisor dof r, the following diagram commutes. Rσd −→ RTd −→ Sd π↓π↓↓π Fprσd −→ FprTd −→ Fpd 2.2. Codes over a chain ring A linear code Cof length nover a ring R, is a submodule of the R-module Rn. We will denote by {0}, the zero-submodule where 0=(0, 0, ..., 0) ∈Rn. A linear code C over Ris free if, C∼ =Rkas R-modules for some positive integer k. The residue code of a linear code Cover Ris the linear code π(C)over Fq, where π(C)={(π(c0),π(c1),··· ,π(cn−1):(c0,c 1,··· ,c n−1)∈C}. In [6], the authors introduced the Galois closure of a linear code Cover Rof length nas follows, Cld(C) =Ext(Td(C)), where Ext(Td(C)) is the linear code over Rof all R-combinations of codewords in the linear code Td(C)over Sd. A linear code Cover R is σd-invariant, if σd(C) =C, where dis a divisor of r. Recall that for any linear code Cover Rof length n, its subring subcode is given by Resd(C) =C∩(Sd)n. In [6], it is shown that any linear code Cover Ris σd-invariant, if and only if, Td(C) =Resd(C) if and only if, C=Ext(Resd(C)). For  ∈{0, 1, ..., r−1}we equip Rnwith the -Galois inner-product defined as follows: u,v= n−1  j=0 ujσ(vj),for all u,v∈Rn. When  =0it is just the usual Euclidean inner-product and if ris even and r=2it is the Hermitian inner-product. The -Galois dual of a linear code Cover Rof length n, denoted C⊥, is defined to be the linear code C⊥={u∈Rn:u,c=0 Rfor all c∈C}. If C⊆C⊥, then Cis -Galois self-orthogonal. Moreover, Cis -Galois self-dual if, C=C⊥. The two statements in Proposition 2below follow immediately from the identity S. Talbi et al. / Finite Fields and Their Applications 77 (2022) 101950 5 u,v=u,σ h(v)−h=σhσ−h(v),ur−h,for all 0 ≤h≤, u,v∈Rn where the action is taken componentwise σ(v) =(σ(v0), ··· , σ(vn−1)). The following proposition is a generalized Delsarte’s Theorem. Proposition 1. ([6, Theorem 3.3]) Let Cbe a linear code over Rof length n. Then for any  ∈{0, 1, ..., r−1}, Td(C⊥) =(Resd(C))⊥. Also [8, Proposition 2.2] has a natural generalization to finite chain rings. Proposition 2. Let Cbe a linear code over Rof length n. Then 1. σh(C)⊥=σh(C⊥), and C⊥=σh(C⊥−h), for any 0 ≤h ≤; 2. (C⊥)⊥h=σ2r−−h(C), for all 0 ≤, h ≤r−1. From Proposition 2and [7, Theorem 3.1], we obtain the following result. Corollary 1. Let Cand Cbe linear codes over Rof length n. Then 1. (C+C)⊥=C⊥∩C⊥ ; 2. (C∩C)⊥=C⊥+C⊥ . Definition 1. Let Cbe a linear code over R. The -Galois hull of Cwill be denoted as H(C), is the intersection of Cand its -Galois dual, that is, H(C)=C∩C⊥. A linear code Cover Ris -Galois Linear Complementary Dual (Shortly, Galois LCD) if H(C) ={0}, and Cis -Galois self-orthogonal if H(C) =C. If we denote that for all 0 ≤; h ≤r−1, we have σh(H(C)) =H(σh(C)), and H(C) =Hr−(C⊥). From the generalized Delsarte’s Theorem in Proposition 1, it follows that Td(H(C)) = (Resd(Hr−(C)))⊥. Note that if Cis  σ-invariant, then H(C) =H0(C). From [14, Proposition 3.2 and Theorem 3.5], for any linear code Cover Rof length n, there is a unique s-tuple (k0, k1, ··· , ks−1)of positive integers, such that Chas a generator matrix in standard form ⎛ ⎜ ⎜ ⎜ ⎝ Ik0G0,1G0,2··· G0,s−2G0,s−1G0,s OθIk1θG1,2··· θG1,s−2θG1,s−1θG1,s ··· ··· ··· ··· ··· ··· ··· OO O··· Oθs−1Iks−1θs−1Gs−1,s ⎞ ⎟ ⎟ ⎟ ⎠U, where Uis a suitable permutation matrix and Othe all zeros matrix of suitable size. The elements in the s-tuple (k0, k1, ··· , ks−1)are called parameters of Cand the rank 6S. Talbi et al. / Finite Fields and Their Applications 77 (2022) 101950 of Cis k0+k1+···+ks−1. From [14, Theorem 3.10], the parameters of C⊥are (n − k, ks−1, ··· , k2, k1), where k=rankR(C). Note that Cis free if and only if rankR(C) = k0and k1=··· =ks−1=0. The q-dimension of a linear code Cover R, denoted dimq(C), is defined to be logq(|C|). Thus the q-dimension of a linear code Cover Rof parameters (k0, k1, ··· , ks−1)is s−1  t=0 (s −t)kt. Since Ris also a Frobenius ring, it follows that dimq(C) +dimq(C⊥) =sn. Proposition 3. Let Cand Cbe two codes over Rof the same length. Then dimq(C+C)=dimq(C)+dimq(C)−dimq(C∩C). Moreover dimq(H(C)) =dimq(Hr−(C)). Proof. The map η:C×C→C+Cdefined as follows: η(x; x) =x +x, is an R-module epimorphism. From the First Isomorphism Theorem, it follows that C×C/Ker(η)and C+Care isomorphic as R-modules. Since Ker(η) ={(x; −x) :x ∈C∩C}, it is easy to see that Ker(η)and C∩Care isomorphic R-modules. Thus |C+C| =|C| |C∩C|×|C|. Therefore logq(|C+C|) =log q(|C|) −logq(|C∩C|) +logq(|C|). From the definition of qdimension of a linear code we have that dimq(C+C) =dimq(C) +dimq(C) −dimq(C∩C). Moreover, dimq(H(C)) = dimq((C+C⊥r−)⊥),from Corollary 1; =sn −dimq(C+C⊥r−),since dimq(C+C⊥r−) +dimq((C+C⊥r−)⊥)=sn; =sn −dimq(C)+dimq(C⊥r−)−dimq(Hr−(C)); =dimq(Hr−(C)),since dimq(C)+dimq(C⊥r−)=sn.  Proposition 4. Let Cbe a free code over Rof length nand be a positive integer. Then 1. dimq(σ(C)) =s ×rank(σ(C)) =s ×dimq(π(σ(C))); 2. π(C)⊥=π(C⊥); 3. π(H(C)) =H(π(C)). Proof. Since Cis free, a generator matrix for σ(C)is Ikσ(A) U, where Ais a k×(n − k)-matrix over Rand Uis a permutation matrix. Thus Ikπ(σ(A)) Uis a generator matrix for π(C). It follows that |σ(C)| =qsk and rank(σ(C)) =dimq(π(σ(C))) =k. This proves Item 1. Now to prove Item 2. The codes π(C)⊥and π(C⊥)have the same parity matrix, which is Ikπ(σ(A)) U. Hence π(C)⊥=π(C⊥). Item 3. is a consequence of the fact that the above diagram commutes, π(H(C)) ⊆H(π(C)) and dimq(π(H(C))) =dimq(H(π(C))).  S. Talbi et al. / Finite Fields and Their Applications 77 (2022) 101950 7 3. Galois hulls of cyclic serial codes Let Nbe the set of nonnegative integers and nbe a positive integer such that gcd(n, q) =1. Set [|a; b|] ={a, a +1, ··· , b}where (a, b) ∈N2such that a <b. Let Aand Bbe two subsets in [|0; n −1|], as usual, the opposite of A, denoted −A, is defined as −A ={n −z:z∈A}and its complementary, denoted A, is defined as: A={z∈[|0; n −1|] :z/∈A}. The set Ais symmetric, if A =−A, and the pair {A, B} is asymmetric, if B =−A. Recall that the pair is a set with two elements. If u ∈N\{0}, then uA ={i∈[|0; n−1|]:(∃z∈A)(uz ≡i(mod n)}. It defines the binary relation on [|0; n −1|]by x ∼qyif there is iin Nsuch that y≡qix(mod n). Obviously, the binary relation ∼qis an equivalence relation on [|0; n −1|]. The cosets of ∼q, are called q-cyclotomic cosets modulo n. Denote by [|0; n −1|]q, a complete system of representatives of ∼q. A subset Zof [|0; n −1|]is a q-closed set modulo n, if Z =qZ. The smallest q-closed set modulo n, containing a subset Zof [|0; n −1|]is i∈NqiZand we will denote it by q(Z). In particular, the set of q-cyclotomic cosets modulo nwhich is q({z}):z∈[|0; n−1|]q, forms a partition of [|0; n −1|]. Since q({z}) ={x ∈[|0; n −1|] :x ∼qz}for any zin [|0; n −1|]. We will take q(∅) =∅by convention. Let jbe a divisor of n, we will use the following notation •φ( . )is the Euler totient function; •ordj(q)the multiplicative order of qmodulo j; •ω(n; q)the number of q-cyclotomic cosets modulo n; •Nq=d∈N\{0}:(∃i∈N\{0})(ddivides qi+1) ; •Λ jthe set of symmetric q-cyclotomic cosets modulo nof size ordj(q); •γ(j; q) := |Λj|; • Λjthe set of asymmetric pairs of q-cyclotomic cosets modulo nof size ordj(q); •β(j; q) := |Λj|. Let δbe a generator of the cyclic multiplicative subgroup Γ(GR(pa, m))\{0}of (GR(pa, m))×, where m =ordn(q). The following result is straightforward from Hensel’s Lemma [12], which guarantees the uniqueness of this monic basic-irreducible factorization of Xn−1, and Xn−1 = z∈[|0;n−1|]q mzwhere mz:=  a∈q({z}) (X−δa). Obviously, for any zin [|0; n −1|]q, the polynomial mzis monic basic-irreducible over R. Lemma 1. The map Ω:q(Z) : Z ⊆[|0; n−1|]q→{f∈GR(pa,r)[X]:fis monic and f|Xn−1} A→  a∈A (X−δa)(1) where Ω(∅) =1, is bijective. Moreover, for any z∈[|0; n −1|]and for all q-closure sets Aand Bmodulo n, we have 8S. Talbi et al. / Finite Fields and Their Applications 77 (2022) 101950 1. Ω q({z})is a monic basic-irreducible polynomial over GR(pa, r)of degree q({z}); 2. lcm (Ω (A) ,Ω(B))=Ω (A ∪B) and gcd (Ω (A) ,Ω(B))=Ω (A ∩B); 3. if A ∩B =∅, then Ω (A ∪B) = Ω (A) Ω (B). Proof. Since δ∈Γ(GR(pa, m))\{0} ⊂GR(pa, m)and GR(pa, m)is a Galois extension of GR(pa, r), it follows that for any q-cyclotomic cosets A modulo n, the monic polynomial  a∈A (X−δa)is basic-irreducible over GR(pa, r). Therefore, the correspondence Ωis well-defined, and by Hensel lemma, Xn−1admits a unique monic basic-irreducible factorization in GR(pa, r)[X]. Thus the existence and the uniqueness of this basic-irreducible factorization over GR(pa, r), the map Ωis bijective. Items 2. and 3. are straightforward to prove.  Proposition 5. [18, Subsection 2.2] Let jbe a divisor of n. Then γ(j;q)=φ(j) ordj(q),if j∈Nq; 0,otherwise, and β(j;q)=φ(j) 2ordj(q),if j/∈Nq, 0,otherwise. Moreover, ω(n; q) = i|n i∈Nq γ(i; q) +2 j|n j/∈Nq β(j; q). We will introduce the following notation En(q,s)=In(q,s)×(Jn(q,s))2,(2) where In(q, s) = i|n i∈Nq Eγ(i;q) sand Jn(q, s) = j|n j/∈Nq Eβ(j;q) s, with Es=(x(0),x (1),··· ,x (s−1))∈{0; 1}s: s−1  a=0 x(a)∈{0; 1}.(3) Note that Es={(0, ··· , 0)} ∪⎧ ⎨ ⎩⎛ ⎝0,··· ,0,1  j-i th position ,0,··· ,0⎞ ⎠:j∈{1; ··· ;s}⎫ ⎬ ⎭⊆ {0; 1}sand |Es| =s +1. The elements in In(q, s)are arrays of the form (((u(a) il )0≤a<s)◦) where (u(a) il )0≤a<s are in Esand the indices iand lsatisfy i | n, i ∈Nqand 1 ≤l≤γ(i; q), i.e., (((u(a) il )0≤a<s)◦)=(u(a) il )0≤a<s1≤l≤γ(i;q)i|n,i∈Nq ∈In(q,s). S. Talbi et al. / Finite Fields and Their Applications 77 (2022) 101950 9 Similarly, (((v(a) jh )0≤a<s)•) =(v(a) jh )0≤a<s1≤h≤β(j;q)j|n,j /∈Nq ∈Jn(q, s). Note that if s =1, then E1={0; 1}, and in this case, we write ((uil)◦) = (((u(a) il )0≤a<1)◦)and ((vjh)•) = (((v(a) jh )0≤a<1)•). Let iand jbe positive integers such that i | n, i ∈Nq, and j| n, j/∈Nq. From now on, Λi={Gil :1≤l≤γ(i;q)}and Λj={{Fjh,−Fjh}:1≤h≤β(j;q)}. Of course, all the polynomials in {Ω(Gil) :1 ≤l≤γ(i; q)}are basic-irreducible in R[X] of degree ordi(q), and all the elements in {{Ω(Fjh), Ω(−Fjh)} :1 ≤h ≤β(j; q)}are pairs of monic basic-irreducible reciprocal polynomials (up to a unit) in R[X]of the same degree ordj(q). The basic-irreducible factorization of Xn−1in R[X]is given as Xn−1=! i|n i∈Nq ⎛ ⎝ γ(i;q) ! l=1 Ω(Gil)⎞ ⎠! j|n j/∈Nq ⎛ ⎝ β(j;q) ! h=1 Ω(Fjh)Ω(−Fjh)⎞ ⎠.(4) Thus, for any monic factor of Xn−1 ∈R[X], there is a unique (((uil)◦), ((vjh)•), ((wjh)•))) in En(q, 1) such that f=! i|n i∈Nq ⎛ ⎝ γ(i;q) ! l=1 Ω(Gil)uil ⎞ ⎠! j|n j/∈Nq ⎛ ⎝ β(j;q) ! h=1 Ω(Fjh)vjh Ω(−Fjh)wjh⎞ ⎠,(5) and conversely. Denote the right-hand side of Equation (5)by ∂(((uil)◦), ((vjh)•), ((wjh)•)). Note that ∂(((1)◦), ((1)•), ((1)•)) =Xn−1and ∂(((0)◦), ((0)•), ((0)•)) =1. If we are given f1=∂(((uil)◦), ((vjh)•), ((wjh)•)) and f2=∂(((u il)◦), ((v jh)•), ((w jh)•)), we have that lcm(f1;f2)=∂(((max{uil,u  il})◦),((max{vjh,v jh})•),((max{wjh,w jh})•)); gcd(f1;f2)=∂(((min{uil,u  il})◦),((min{vjh,v jh})•),((min{wjh,w jh})•)), and if all (uil +u il, vjh +v jh, wjh +w jh)are in {0; 1}3then f1f2=∂(((uil +u il)◦),((vjh +v jh)•),((wjh +w jh)•)).(6) A cyclic code Cof length nover Ris a linear code that is invariant under the transformation τ((c0, c1, ··· , cn−1)) =(cn−1, c0, ··· , cn−2). If we denote by Xn−1the ideal of R[X] generated by Xn−1, it is well-known that any cyclic code of length nover R can be represented as an ideal of the quotient ring R[X]/Xn−1via the R-module isomorphism Ψ:Rn→R[X]/Xn−1, where Ψ(c) =Ψ(c) +Xn−1and 16 S. Talbi et al. / Finite Fields and Their Applications 77 (2022) 101950 1. Cis LCD if, and only if x(2) il =x(1) il , y(2) jh =y(1) jh , z(2) jh =z(1) jh and (x(2) il ; y(2) jh ; z(2) jh ) ∈ {(0; 0; 0), (0; 1; 1), (1; 0; 0), (1; 1; 1)}, for all i, l, j, h. 2. Cis self-orthogonal if, and only if (x(2) il ; x(1) il ) ∈{(0; 1), (1; 0)}and (y(2) jh ;y(1) jh ;z(2) jh ;z(1) jh )∈{(1;0;1;0),(0;1;1;0),(1;0;0;1),(1;0;0;0),(0;0;1;0), (0;1;0;1)}, for all i, l, j, h. Note that Corollary 3is insufficient to characterize the nontrivial self-dual cyclic codes over Rwhen sis even (see [4, Theorem 4.4]). 4. The q-dimensions of Euclidean hulls of cyclic serial codes In this section, Cis a cyclic serial code of length nover Rwith triple-sequence (((x(a) il )0≤a<s)◦),(((y(a) jh )0≤a<s)•),(((z(a) jh )0≤a<s)•) in En(q, s). Then Ψ(C)=(θt·∂+++ s  a=t+1 x(a) il ,◦,,++ s  a=t+1 y(a) jh ,•,,++ s  a=t+1 z(a) jh ,•,,: 0≤t≤s−1). From Corollary 3, Ψ(H0(C)) = (θt·∂+++ s  a=t+1 u(a) il ,◦,,++ s  a=t+1 v(a) jh ,◦,,++ s  a=t+1 w(a) jh ,◦,,: 0≤t≤s−1), where ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ s  a=t+1 u(a) il =1−min .t  a=0 x(a) il ;1− s−t−1  a=0 x(a) il /; s  a=t+1 v(a) jh =1−min .t  a=0 y(a) jh ;1− s−t−1  a=0 z(a) jh /; s  a=t+1 w(a) jh =1−min .t  a=0 z(a) jh ;1− s−t−1  a=0 y(a) jh /, S. Talbi et al. / Finite Fields and Their Applications 77 (2022) 101950 17 for all 0 ≤t ≤s −1. The following notations are important for the sequel of this paper. For all 0 ≤t ≤s −1, 1 ≤l≤γ(i; q)and 1 ≤h ≤β(j; q), denote by: ε(t) jh = s  a=t+1 (v(a) jh +w(a) jh ).(12) Note that ε(−1) jh =2. Let us consider now il= s−1  t=0 (s−t)u(t) il ,and jh = s−1  t=0 (s−t)(ε(t−1) il −ε(t) il ).(13) Obviously, il= s−1  t=0 (t) il , where (t) il =min .t  a=0 x(a) il ;1− s−t−1  a=0 x(a) il /, and jh = s−1  t=0 (t) jh, where (t) jh =mint  a=0 y(a) jh ;1− s−t−1  a=0 z(a) jh +mint  a=0 z(a) jh ;1− s−t−1  a=0 y(a) jh . Thus, we set i:= γ(i;q)  l=1 il, ε(t) j:= β(j;q)  h=1 ε(t) jh and j:= β(j;q)  h=1 jh. Remark 3. Let 0 ≤t ≤s −1. 1. (t) il ∈{0; 1}and (t) jh ∈{0; 1; 2}. 2. If 0 <t <s, then (t−1) il ≤(t) il and (t−1) jh ≤(t) jh. 3. If 2t <s, then (t) il =0and (t) jh ≤1. Lemma 4. Let jbe a divisor of nsuch that j/∈Nq. Then 0≤ε(t−1) j−ε(t) j≤β(j;q)−(ε(t−2) j−ε(t−1) j),if t<2s 23; 0≤ε(t−1) j−ε(t) j≤2β(j;q)−(ε(t−2) j−ε(t−1) j),if t≥2s 23. Proof. Let 0 ≤t ≤s −1and (t) j= β(j;q)  h=1 (t) jh. We have ε(t) jh =2 −(t) jh. From Remark 3, two cases are considered. Let (t−1) j:= |{h ∈N:1 ≤h ≤β(j; q)andε(t−1) jh = ε(t) jh =1}|. Then there is a permutation τin Sβ(j;q)such that ε(t−1) jh =ε(t) jh =1, for all h ∈{τ(1), ··· , τ((t−1) j)}. Obviously, ε(t−2) j≤2β(j; q). For that ε(t−2) j−ε(t−1) j≤(t−1) j. 18 S. Talbi et al. / Finite Fields and Their Applications 77 (2022) 101950 Case 1: t<2s 23.We have ε(t) jh ∈{1; 2}, and ε(t−1) jh −ε(t) jh ∈{0; 1},if ε(t−1) jh =2; ε(t) jh =ε(t−1),if ε(t−1) jh =1. Thus ε(t−1) j−ε(t) j=⎛ ⎜ ⎝ h∈{τ(1),··· ,τ((t−1) j)} (ε(t−1) jh −ε(t) jh)⎞ ⎟ ⎠ +⎛ ⎜ ⎝ h∈{τ((t−1) j+1),··· ,τ(β(j;q))} (ε(t−1) jh −ε(t) jh)⎞ ⎟ ⎠; =0+⎛ ⎜ ⎝ h∈{τ((t−1) j+1),··· ,τ(β(j;q))} (ε(t−1) jh −ε(t) jh)⎞ ⎟ ⎠, since 0 ≤ε(t−1) jh −ε(t) jh ≤1. Hence 0 ≤ε(t−1) j−ε(t) j≤β(j; q) −(t−1) j≤β(j; q) −(ε(t−2) j−ε(t−1) j). Case 2: t≥2s 23.We have ε(t−1) jh −ε(t) jh ∈{0; 1; 2},if ε(t−1) jh ∈{1; 2}; ε(t) jh =ε(t−1) jh ,if ε(t−1) jh =0. Thus ε(t−1) j−ε(t) j=⎛ ⎜ ⎝ h∈{τ(1),··· ,τ((t−1) j)} (ε(t−1) jh −ε(t) jh)⎞ ⎟ ⎠ +⎛ ⎜ ⎝ h∈{τ((t−1) j+1),··· ,τ(β(j;q))} (ε(t−1) jh −ε(t) jh)⎞ ⎟ ⎠; =0+⎛ ⎜ ⎝ h∈{τ((t−1) j+1),··· ,τ(β(j;q))} (ε(t−1) jh −ε(t) jh)⎞ ⎟ ⎠, since 0 ≤ε(t−1) jh −ε(t) jh ≤2. Therefore 0 ≤ε(t−1) j−ε(t) j≤2(β(j; q) −(t−1) j) ≤2 β(j;q)−(ε(t−2) j−ε(t−1) j). Theorem 2. The parameters of the Euclidean hull of a cyclic serial code over Rof length nare given by (k0, k1, ··· , ks−1)where 2k0+k1+···+ks−1≤n, kt= i|n i∈Nq ordi(q)·u(t) i+ j|n i/∈Nq ordj(q)·ν(t) j, S. Talbi et al. / Finite Fields and Their Applications 77 (2022) 101950 19 with u(t) i=0,if t<2s 23; 0≤u(t) i≤γ(i;q),if t≥2s 23,and ⎧ ⎪ ⎨ ⎪ ⎩ ε(t) j=0,if n∈Nq; 0≤ν(t) j≤β(j;q)−ν(t−1) j,if n/∈Nq,andt<2s 23; 0≤ν(t) j≤2(β(j;q)−ν(t−1) j),if n/∈Nq,andt≥2s 23. Moreover ν(−1) j=0. Proof. Let (k0, k1, ··· , ks−1)be the parameters of H0(C). When H0(C) =C, we have 2k0+k1+···+ks−1≤n. Then for all 0 ≤t ≤s −1, kt=deg +∂+++s  a=t u(a) il ,◦,,++s  a=t v(a) ij ,◦,,++s  a=t w(a) jh ,◦,,, −deg +∂+++ s  a=t+1 u(a) il ,◦,,++ s  a=t+1 v(a) jh ,◦,,++ s  a=t+1 w(a) jh ,◦,,,; = i|n i∈Nq ordi(q)·u(t) i+ j|n i/∈Nq ordj(q)·(ε(t−1) j−ε(t) j),where u(t) i= γ(i;q)  l=1 u(t) il . Since u(t) i=0,if t<2s 23; 0≤u(t) i≤γ(i;q),if t≥2s 23, it follows that u(t) i=0,if 2t<s; 0≤u(t) i≤γ(i;q),if s≤2t. On the other hand, one notes that if n ∈Nq, then any positive divisor of nis in then Nq. By Lemma 4, we obtain ⎧ ⎪ ⎨ ⎪ ⎩ ε(t) j=0,if n∈Nq; 0≤ν(t) j≤β(j;q)−ν(t−1) j,if n/∈Nq,andt<2s 23; 0≤ν(t) j≤2(β(j;q)−ν(t−1) j),if n/∈Nq,andt≥2s 23, where ν(t) j=ε(t−1) j−ε(t) j. Obviously ν(−1) j=ε(−2) j−ε(−1) j=0.  The previous discussion leads to the Algorithm 1and justifies its correctness. Examples 4.1, 4.2, 4.3 show different outputs of the algorithm. Example 4.1. All possible parameters of Euclidean hulls of cyclic codes of length 11 over Z27 are determined as follows. 1. The divisors of 11 are 1and 11. a) We have 1 ∈N3, so ord1(3) =1and γ(1; 3) =1. 20 S. Talbi et al. / Finite Fields and Their Applications 77 (2022) 101950 Algorithm 1: Parameters of the Euclidean hull of a cyclic serial code over R. Input: Length n, and a finite chain ring Rof parameters (p, a, r, e, s)such that gcd(p, n) =1. Output: All possible s-tuples (k0, k1, ··· , ks−1) describing the parameters of the Euclidean hull of a cyclic serial code 1.if n ∈Nqthen 2for 0 ≤t <sdo 3if t <2s 23then 4kt=0. 5else 6For each i | n, compute ordi(q), and γ(i; q), 7therefore all the possible values of kt, such that kt= i|n i∈Nq ordi(q)·u(t) i, with 0 ≤u(t) i≤γ(i; q). 8return The possible parameters (0, ··· , 0, k4s 25, ··· , ks−1)such that k4s 25+···+ks−1≤n. 9else 10 For each i | n, if i ∈Nq, then compute ordi(q), and γ(i; q). 11 For each j| n, if j/∈Nq, then compute ordj(q), and β(j; q). 12 for 0 ≤t <s,do 13 if t =0then 14 compute k0= j|n i/∈Nq ordj(q) ·ν(0) j, where 0 ≤ν(0) j≤β(j; q) 15 else 16 while 0 <t <2s 23do 17 For a fixed ν(t−1) jin kt−1, compute kt= j|n i/∈Nq ordj(q) ·ν(t) j, where 0 ≤ν(t) j≤β(j; q) −ν(t−1) j, 18 if 2k0+k1+···+kt≤nthen 19 consider kt, 20 else 21 reject kt 22 while t ≥2s 23do 23 For a fixed ν(t−1) jin kt−1, compute kt= i|n i∈Nq ordi(q) ·u(t) i+ j|n i/∈Nq ordj(q) ·ν(t) j, where 0 ≤u(t) i≤γ(i; q)and 0 ≤ν(t) j≤2 ·(β(j; q) −ν(t−1) j). 24 if 2k0+k1+···+kt≤nthen 25 consider kt, 26 else 27 reject kt 28 return The possible parameters (k0, k1, ··· , ks−1)describing the Euclidean hull of a cyclic serial code. b) We have 11 /∈N3, so ord11(3) =5and β(11; 3) =1. 2. It follows that k0=5ν(0) 11 ,where 0 ≤ν(0) 11 ≤1 k1=5ν(1) 11 ,where 0 ≤ν(1) 11 ≤1−ν(0) 11 k2=u(2) 1+5ν(2) 11 where 0 ≤u(2) 1≤1and0≤ν(2) 11 ≤2(1 −ν(1) 11 ). S. Talbi et al. / Finite Fields and Their Applications 77 (2022) 101950 21 Hence, the all possible parameters (k0, k1, k2)of the Euclidean hulls of cyclic codes of length 7over Z8are given in the following table k0k1k2 0 0 0, 1, 5, 6, 10, 11 50,1 500,1 Example 4.2. All the possible parameters (k0, k1, k2)of the Euclidean hull of a cyclic code of length 7over Z8are determined as follows. 1. The divisors of 7are 1and 7. a) We have 1 ∈N2, so ord1(2) =1and γ(1; 2) =1. b) We have 7 /∈N2, so ord7(2) =3and β(7; 2) =1. 2. It follows that k0=3ν(0) 7,where 0 ≤ν(0) 7≤1 k1=3ν(1) 7,where 0 ≤ν(1) 7≤1−ν(0) 7 k2=u(2) 1+3ν(2) 7where 0 ≤u(2) 1≤1and0≤ν(2) 7≤2(1 −ν(1) 7). Hence, the all possible parameters (k0, k1, k2)of the Euclidean hulls of cyclic codes of length 7over Z8are given in the following table k0k1k2 0 0 0, 1, 3, 4, 6, 7 30,1 300,1 Example 4.3. The parameters of the Euclidean hulls of cyclic codes of length 21 over Z8 are given by 1. The divisors of 21 are {1, 3, 7, 21}. (a) 1; 3 ∈N2, we have ord1(2) =1, ord3(2) =2and γ(1; 2) =γ(3; 2) =1. (b) 7; 21 /∈N2, we have ord7(2) =3, ord21(2) =6and β(7; 2) =β(21; 2) =1. 2. It follows that k0=3ν(0) 7+6ν(0) 21 ,with 0 ≤ν(0) j≤1,where j∈{7; 21}. k1=3ν(1) 7+6ν(1) 21 ,with 0 ≤ν(1) j≤1−ν(0) j,where j∈{7; 21}. k2=u(2) 1+2u(2) 3+3ν(2) 7+6ν(2) 21 ,with 0 ≤u(2) i≤1and0≤ν(2) j≤2(1 −ν(1) j), where i∈{1; 3},and j∈{7; 21}. Hence, the all possible parameters (k0, k1, k2)of the Euclidean hulls of cyclic codes of length 21 over Z8are given in the following table 22 S. Talbi et al. / Finite Fields and Their Applications 77 (2022) 101950 k0k1k2 0 0 0, 1, 2, 3, ···,21 3 0, 1, 2, 3, 6, 7, 8, 9, 12, 13, 14, 15 6 0,1,2,3,4,5,6,7,8,9 9 0,1,2,3 3 0 0, 1, 3, ···,15 6 0,1,2,3,4,5,6,7,8,9 6 0 0, 1, 3, ···,9 3 0, 1, 2, 3, 6, 7, 8, 9, 12, 13, 14, 15 9 0 0, 1, 2, 3 Corollary 4. The set ℵ(n, s, q)of q-dimensions of the Euclidean hull of a cyclic serial code of length nover R, is given by ℵ(n, s, q)=⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ i|n i∈Nq ordi(q)⎛ ⎝ γ(i;q)  l=1 il⎞ ⎠+ j|n i/∈Nq ordj(q)⎛ ⎝ β(j;q)  h=1 jh⎞ ⎠|0≤il≤s−2s 23 0≤jh ≤s⎫ ⎪ ⎪ ⎬ ⎪ ⎪ ⎭ . Proof. Let Cbe a cyclic serial code of length nover Rwith triple-sequence (((x(a) il )0≤a<s)◦),(((y(a) jh )0≤a<s)•),(((z(a) jh )0≤a<s)•) in En(q, s). From Theorem 2, the parameters (k0, k1, ··· , ks−1)of H0(C) where for all 0 ≤t ≤s −1, kt= i|n i∈Nq ordi(q)·⎛ ⎝ γ(i;q)  i=1 u(t) il ⎞ ⎠+ j|n i/∈Nq ordj(q)·⎛ ⎝ β(j;q)  h=1 (ε(t−1) jh −ε(t) jh)⎞ ⎠. Thus the q-dimension of H0(C)is s−1  t=0 (s −t)kt. It follows that dimq(C)= i|n i∈Nq ordi(q)·⎛ ⎝ γ(i;q)  i=1 il⎞ ⎠+ j|n i/∈Nq ordj(q)·⎛ ⎝ β(j;q)  h=1 jh⎞ ⎠. From Remark 3, il= s−1  t=0 (t) il = s−1  t=2s 23 (t) il ≤s−4s 25, and if j∈Nqthen j=0. Otherwise, jh = s−1  t=0 (t) jh =2s 23−1  t=0 (t) jh + s−1  t=2s 23 (t) jh S. Talbi et al. / Finite Fields and Their Applications 77 (2022) 101950 23 ≤max 0≤b≤s−2s 23#4s 25+b+2s−4s 25−b$=s.  5. The average q-dimension We will denote by C(n; R)the set of all cyclic serial codes over length nover R. The average q-dimension of the Euclidean hull of cyclic of length nover Ris ER(n)=  C∈C(n;R) dimq(H0(C)) |C(n;R)|. In this section, an explicit formula for ER(n)and bounds are given in terms of Bn,q where Bn,q =deg ! i|n i∈Nq ⎛ ⎝ γ(i;q) ! l=1 Ω(Gil)⎞ ⎠= i|n i∈Nq φ(i), where Gil are symmetric q-cyclotomic cosets modulo nof size ordj(q), as defined in (4). Consider the maps :Es→N (x(0),··· ,x (s−1))→ s−1  t=0 min .t  a=0 x(a);1− s−t−1  a=0 x(a)/,(14) and  :Es×Es→Ndefined as (y,z)= s−1  t=0 +min t  a=0 y(a);1− s−t−1  a=0 z(a)+mint  a=0 z(a);1− s−t−1  a=0 y(a),, (15) where (y, z) =((y(0), ··· , y(s−1)), (z(0), ··· , z(s−1))). Let τ∈ℵ(n, s, q)be an element in the set defined in Corollary 4. Then τis the qdimension of the Euclidean hull of a cyclic serial code of length nover R. The following result gives the number of cyclic serial codes of length nover Rwhose Euclidean hulls have q-dimension τ. Proposition 8. Let nbe a positive integer such that gcd(n, p) =1and τ∈ℵ(n, s, q) where ℵ(n, s, q)is described in Corollary 4. The number ℘(n, τ; R)of cyclic serial codes of length nover Rwhose Euclidean hulls have q-dimension τis given by: ℘(n, τ;R)=  (((il)◦),((jh)•))∈Υ(τ) ⎛ ⎜ ⎜ ⎝! i|n i∈Nq γ(i;q) ! l=1 ψs(il)⎞ ⎟ ⎟ ⎠⎛ ⎜ ⎜ ⎝! j|n j/∈Nq β(j;q) ! h=1 ρs(jh)⎞ ⎟ ⎟ ⎠, 24 S. Talbi et al. / Finite Fields and Their Applications 77 (2022) 101950 where ψs(il)=|{x∈Es:(x)=il}|,ρ s(jh)=|{(y,z)∈Es×Es:(y,z)=jh}|, and Υ(τ)=⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ (((il)◦),((jh)•)) :  i|n i∈Nq ordi(q)⎛ ⎝ γ(i;q)  l=1 il⎞ ⎠+ j|n i/∈Nq ordj(q)⎛ ⎝ β(j;q)  h=1 jh⎞ ⎠=τ⎫ ⎪ ⎪ ⎬ ⎪ ⎪ ⎭ . The above expression of ER(n) = τ∈ℵ(n,s,q) τ·℘(n,τ;R) |C∈C(n;R)|, might lead to a tedious and lengthy computation. The remainder of the section will show an alternative simpler expression for the expected value. Lemma 5. Consider the random variable defined in (14)with uniform probability. The expected value E()is given by: E()=2s 23s−2s 23 s+1 =s2 4(s+1) ,if seven; s−1 4,if sodd. Proof. Let t ∈{0; 1; ··· ; s −1}and x=(x(0), ··· , x(s−1)) ∈Es. Set (t) (x)=mint  a=0 x(a);1− s−t−1  a=0 x(a)∈{0; 1}. Then (t) (x)=1if and only if 2t ≥sand , t  a=s−t x(a) il =1. Thus for all η∈N, we have |{x∈Es:(t) (x)=η}| =2t−s+1,if t≥2s 23and η=1; 0,otherwise. Therefore, |{x∈Es:(x)=η}| =⎧ ⎪ ⎨ ⎪ ⎩ s−1  t=2s 23(2t−s+1),if η=s−2s 23; 0,otherwise. =2s 23s−2s 23,if η=s−2s 23; 0,otherwise. Since |Es| =s +1and P({x∈Es:(x) =η}) =|{(x)=η}| |Es|, it follows that, E()= η∈N ηP({x∈Es:(x)=η})=2s 23s−2s 23 s+1 . S. Talbi et al. / Finite Fields and Their Applications 77 (2022) 101950 25 Lemma 6. Consider the random variable  :Es×Es→Ndefined in (15)with uniform distribution. The expected value E()is given by E()=s(2s+1) 3(s+1). Proof. From Corollary 4, for any (y, z) ∈Es×Es, 0 ≤(y, z) ≤s. Let Es(η)={(y,z)∈Es×Es:(y,z)=η}, for 0 ≤η≤s. Now, |Es(η)|=2(η+1),if 0 ≤η≤s−1; s+1,if η=s. Thus E()= 1 (s+1) 2 s  η=0 η|Es(η)|; =1 (s+1) 2+s−1  η=1 2η(η+1)+s(s+1) ,; =s(2s2+3s+1) 3(s+1) 2. Theorem 3. The average q-dimension of the Euclidean hull of cyclic serial codes from C(n; R)is ER(n)=⎧ ⎨ ⎩(2s+1)s 6(s+1) n−(s+2)s 12(s+1) Bn,q,if seven; (2s+1)s 6(s+1) n−s2+2s+3 12(s+1) Bn,q,if sodd, where Bn,q = i|n i∈Nq φ(i). Proof. Let Ybe the random variable that takes as value dimq(H0(C)) when we choose at random a cyclic serial code from C(n; R)with uniform probability. Then E(Y) =ER(n). By Lemma 3, there exists an one-to-one correspondence between C(n; R), and En(q, s). Therefore, choosing a cyclic serial code Cfrom C(n, R)their probabilities are identical. By Corollary 4, we obtain Y= i|n i∈Nq ordi(q)⎛ ⎝ γ(i;q)  l=1 il⎞ ⎠+ j|n i/∈Nq ordj(q)⎛ ⎝ β(j;q)  h=1 jh⎞ ⎠.