ISSN 2148-838Xhttps://doi.org/10.13069/jacodesmath.935951 J. Algebra Comb. Discrete Appl. 8(2) • 91–105 Received: 20 May 2020 Accepted: 8 December 2020 Journal of Algebra Combinatorics Discrete Structures and Applications Composite G-codes over formal power series rings and finite chain rings Research Article Adrian Korban Abstract: In this paper, we extend the work done on G-codes over formal power series rings and finite chain rings Fq[t]/(ti), to composite G-codes over the same alphabets. We define composite G-codes over the infinite ring R∞ as ideals in the group ring R∞G. We show that the dual of a composite G-code is again a composite G-code in this setting. We extend the known results on projections and lifts of G-codes over the finite chain rings and over the formal power series rings to composite G-codes. Additionally, we extend some known results on γ-adic G-codes over R∞ to composite G-codes and study these codes over principal ideal rings. 2010 MSC: 94B05, 16S34 Keywords: Composite G-codes, Group rings, Finite chain rings, Formal power series rings, p-adic integers 1. Introduction In [11], T. Hurley introduced a map σ which sends the group ring element v ∈ RG to a matrix σ(v) over the ring R. The author also used this map to construct and study codes over fields. The feature of this map is that for different finite groups in the group ring element v, the map σ(v) will produce different matrices over the ring R. For example in [10], the authors show that if v ∈ RD2n then the generator matrix of the form [In | σ(v)] produces the well-known four circulant construction used in coding theory. In [7], the authors apply the above map and study codes generated by 〈σ(v)〉 over the Frobenius rings. They define G-codes which are ideals in the group ring RG, where R is a finite commutative Frobenius ring and G is a finite group. In [4], the authors study G-codes over formal power series rings and finite chain rings. They extend many well known results on codes over Ri and R∞ to G-codes over the same alphabets. The authors also study γ-adic G-codes over R∞ and G-codes over principal ideal rings. Adrian Korban; Department of Mathematical and Physical Sciences, University of Chester, Thornton Science Park, Pool Ln, Chester CH2 4NU, England (email: adrian3@windowslive.com). 91 https://orcid.org/0000-0001-5206-6480 A. Korban / J. Algebra Comb. Discrete Appl. 8(2) (2021) 91–105 Recently in [3], the authors extended the map σ introduced by T. Hurley in [11], so that the group ring element v gets sent to more complex matrices over the ring R. The authors denote this map Ω and call the matrices Ω(v) the composite matrices- see [3] for details. In [6], the authors introduce and study composite G-codes which are defined by taking the row space of the composite matrix Ω(v), i.e., 〈Ω(v)〉. They also extend many results from [4] on G-codes to composite G-codes. In this work, we generalize the results on G-codes over formal power series rings and finite chain rings Fq[t]/(ti) from [4] and some results from [8] to composite G-codes over the same alphabets. We study the projections and lifts of composite G-codes over the finite chain rings and over the formal power series rings respectively. We also extend the results on γ-adic G-codes over R∞ to composite G-codes and some results on G-codes over principal ideal rings to composite G-codes. In many parts of this work, the results we present are a simple generalization or a consequence of the results proven in [4] and [8]. The rest of the work is organized as follows. In Section 2, we give preliminary definitions and results on codes, finite chain rings, formal power series and composite G-codes. In Section 3, we show that the composite G-codes are ideals in the group ring R∞G. In Section 4, we study the projections and lifts of the composite G-codes with a given type. In Sections 5 and 6, we extend the results from [4]; we study self-dual γ-adic composite G-codes and composite G-codes over principal ideal rings. We finish with concluding remarks and directions for possible future research. 2. Preliminaries 2.1. Codes We shall give the definitions for codes over rings. For a complete description of algebraic coding theory in this setting, see [2]. Let R be a commutative ring. A code of length n over R is a subset of Rn and a code is linear if it is a submodule of the ambient space Rn. We assume that all finite rings we use as alphabets are Frobenius, where a Frobenius ring is characterized by the following. Let R̂ be the character module of the ring R. For a finite ring R the following are equivalent: • R is a Frobenius ring. • As a left module, R̂ ∼= RR. • As a right module, R̂ ∼= RR. The Hamming weight of a vector is the number of non-zero coordinates in that vector and the minimum weight of a code is the smallest weight of all non-zero vectors in the code. We define the standard inner-product on the ambient space, namely [v,w] = ∑ viwi. We define the orthogonal with respect to this inner-product as: C⊥ = {v ∈ Rn | [v,w] = 0,∀w ∈C}. The code C⊥ is linear, whether or not C is. If R is a finite Frobenius ring, then we have that (C⊥)⊥ = C for all linear codes C over R. However, if R is infinite this is not always true. Definition 2.1. A linear code C over an infinite ring R is called basic if C = (C⊥)⊥. 2.2. Finite chain rings and formal power series rings We recall the definitions and properties of a finite chain ring R and the formal power series ring R∞. We refer the reader to [8] and [9] for details and further explanations. In this paper, we assume that all 92 A. Korban / J. Algebra Comb. Discrete Appl. 8(2) (2021) 91–105 rings have a multiplicative identity and that all rings are commutative. We also stress that the results we present in this work are given only for finite chain rings Fq[t]/(ti). 2.2.1. Finite chain rings A ring is called a chain ring if its ideals are linearly ordered by inclusion. In particular, this means that any finite chain ring has a unique maximal ideal. Let R be a finite chain ring. Denote the unique maximal ideal of R by m, and let γ̃ be the generator of the unique maximal ideal m. This gives that m = 〈γ̃〉 = Rγ̃, where Rγ̃ = 〈γ̃〉 = {βγ̃ | β ∈ R}. We have the following chain of ideals: R = 〈γ̃0〉⊇ 〈γ̃1〉⊇ ·· · ⊇ 〈γ̃i〉⊇ ·· · . (1) The chain in (1) can not be infinite, since R is finite. Therefore, there exists i such that 〈γ̃i〉 = {0}. Let e be the minimal number such that 〈γ̃e〉 = {0}. The number e is called the nilpotency index of γ̃. This gives that for a finite chain ring we have the following: R = 〈γ̃0〉⊇ 〈γ̃1〉⊇ ·· · ⊇ 〈γ̃e〉. (2) If the ring R is infinite then the chain in Equation 1 is also infinite. Let R× denote the multiplicative group of all units in the ring R. Let F = R/m = R/〈γ̃〉 be the residue field with characteristic p, where p is a prime number, then |F| = q = pr for some integers q and r. We know that |F×| = pr −1. We now state two well-known lemmas for which the proofs can be found in [12]. Lemma 2.2. For any 0 6= r ∈ R there is a unique integer i, 0 ≤ i < e such that r = µγ̃i, with ν a unit. The unit µ is unique modulo γ̃e−i. Lemma 2.3. Let R be a finite chain ring with maximal ideal m = 〈γ̃〉, where γ̃ is a generator of m with nilpotency index e. Let V ⊆ R be a set of representatives for the equivalence classes of R under congruence modulo γ̃. Then (i) for all r ∈ R there are unique r0, · · · ,re−1 ∈ V such that r = ∑e−1 i=0 riγ̃ i; (ii) |V | = |F|; (iii) |〈γ̃j〉| = |F|r−j for 0 ≤ j ≤ e− 1. From Lemma 2.3, we know that any element ã of R can be written uniquely as ã = a0 + a1γ̃ + · · · + ae−1γ̃e−1, where the ai can be viewed as elements in the field F. It is well-known that the generator matrix for a code C over a finite chain ring Ri, where i < ∞ is permutation equivalent to a matrix of the following form: G =   Ik0 A0,1 A0,2 A0,3 A0,e γIk1 γA1,2 γA1,3 γA1,e γ2Ik2 γ 2A2,3 γ 2A2,e ... ... ... ... γe−1Ike−1 γ e−1Ae−1,e   , (3) where e is the nilpotency index of γ. This matrix G is called the standard generator matrix form for the code C. In this case, the code C is said to have type 1k0γk1 (γ2)k2 . . . (γe−1)ke−1. (4) 93 A. Korban / J. Algebra Comb. Discrete Appl. 8(2) (2021) 91–105 2.2.2. Formal power series rings In the next definitions, which can be found in [8], γ will indicate the generator of the ideal of a chain ring, not necessarily the maximal ideal. Definition 2.4. The ring R∞ is defined as a formal power series ring: R∞ = F[[γ]] = { ∞∑ l=0 alγ l|al ∈ F}. Let i be an arbitrary positive integer. The rings Ri are defined as follows: Ri = {a0 + a1γ + · · · + ai−1γi−1|ai ∈ F}, where γi−1 6= 0, but γi = 0 in Ri. If i is finite or infinite then the operations over Ri are defined as follows: i−1∑ l=0 alγ l + i−1∑ l=0 blγ l = i−1∑ l=0 (al + bl)γ l (5) i−1∑ l=0 alγ l · i−1∑ l′=0 bl′γ l′ = i−1∑ s=0 ( ∑ l+l′=s albl′)γ s. (6) The following results can be found in [8]. 1. The ring Ri is a chain ring with the maximal ideal 〈γ〉 for all i < ∞. 2. The multiplicative group R×∞ = { ∑∞ j=0 ajγ j|a0 6= 0}. 3. The ring R∞ is a principal ideal domain. Let C be a finitely generated linear code over R∞. Then the generator matrix of code C is permutation equivalent to the following standard form generator matrix. Let C be a finitely generated, nonzero linear code over R∞ of length n, then any generator matrix of C is permutation equivalent to a matrix of the following form: G =   γm0Ik0 γ m0A0,1 γ m0A0,2 γ m0A0,3 γ m0A0,r γm1Ik1 γ m1A1,2 γ m1A1,3 γ m1A1,r γm2Ik2 γ m2A2,3 γ m2A2,r ... ... ... ... γmr−1Ikr−1 γ mr−1Ar−1,r   , (7) where 0 ≤ m0 < m1 < · · · < mr−1 for some integer r. The column blocks have sizes k0,k1, . . . ,kr and ki are nonnegative integers adding to n. Definition 2.5. A code C with generator matrix of the form given in Equation 7 is said to be of type (γm0 )k0 (γm1 )k1 . . . (γmr−1 )kr−1, where k = k0 + k1 + · · · + kr−1 is called its rank and kr = n−k. A code C of length n with rank k over R∞ is called a γ-adic [n,k] code. We call k the dimension of C and we write by dim C = k. 94 A. Korban / J. Algebra Comb. Discrete Appl. 8(2) (2021) 91–105 Let i,j be two integers with i ≤ j, we define a map Ψ j i : Rj → Ri, (8) j−1∑ l=0 alγ l 7→ i−1∑ l=0 alγ l. (9) If we replace Rj with R∞ then we obtain a map Ψ∞i . For convenience, we denote it by Ψi. It is easy to get that Ψji and Ψi are ring homomorphisms. Let a,b be two arbitrary elements in Rj. It is easy to get that Ψ j i (a + b) = Ψ j i (a) + Ψ j i (b), Ψ j i (ab) = Ψ j i (a)Ψ j i (b). (10) If a,b ∈ R∞, we have that Ψi(a + b) = Ψi(a) + Ψi(b), Ψi(ab) = Ψi(a)Ψi(b). (11) Note that the map Ψji and Ψi can be extended naturally from R n j to R n i and R n ∞ to R n i . The construction method above gives a chain of rings where Ri is a finite ring for all finite i and R∞ is an infinite principal ideal domain. This gives the following diagram: R F ‖ ‖ R∞ → ··· → Re → Re−1 → ··· → R1 2.3. Composite G-codes In this section, we define a circulant matrix, give the definitions for group rings and introduce composite G- codes. A circulant matrix is one where each row is shifted one element to the right relative to the preceding row. We label the circulant matrix as A = circ(α1,α2, . . . ,αn), where αi are ring elements. We shall now give the necessary definitions for group rings. Let G be a finite group of order n and let R be a ring, then the group ring RG consists of ∑n i=1 αigi, αi ∈ R, gi ∈ G. Addition in the group ring is done by coordinate addition, namely n∑ i=1 αigi + n∑ i=1 βigi = n∑ i=1 (αi + βi)gi. (12) The product of two elements in a group ring is given by ( n∑ i=1 αigi)( n∑ j=1 βjgj) = ∑ i,j αiβjgigj. (13) It follows that the coefficient of gk in the product is ∑ gigj=gk αiβj. The following matrix construction was first introduced in [3]. In [6], the authors have shown that the same construction produces codes in Rn from elements in the group ring RG. Let {g1,g2, . . . ,gn} be a fixed listing of the elements of G. Let {h1,h2, . . . ,hr} be a fixed listing of the elements of H, where H is a group of order r. Here, let r be a factor of n with n > r and n,r 6= 1. Also, let Gr be a subset of G containing r distinct elements of G. Define the map: 95 A. Korban / J. Algebra Comb. Discrete Appl. 8(2) (2021) 91–105 φ : H 7→ Gr h1 φ−→ g1 h2 φ−→ g2 ... ... ... hr φ−→ gr. Next, let v = αg1g1 + αg2g2 + · · · + αgngn ∈ RG. Define the matrix Ω(v) ∈ Mn(R) to be Ω(v) =   A1 A2 A3 . . . An r An r +1 An r +2 An r +3 . . . A2n r ... ... ... ... ... A(r−1)n r +1 A(r−1)n r +2 A(r−1)n r +3 . . . An2 r2   , (14) where at least one block has the following form: A′l =   αg−1 j gk αg−1 j gk+1 . . . αg−1 j gk+(r−1) αφl((hl)−12 (hl)1) αφl((hl)−12 (hl)2) . . . αφl((hl)−12 (hl)r) αφl((hl)−13 (hl)1) αφl((hl)−13 (hl)2) . . . αφl((hl)−13 (hl)r) ... ... ... ... αφl((hl)−1r (hl)1) αφl((hl)−1r (hl)2) . . . αφl((hl)−1r (hl)r)   , and the other blocks are of the form: Al =   αg−1 j gk αg−1 j gk+1 . . . αg−1 j gk+(r−1) αg−1 j+1 gk αg−1 j+1 gk+1 . . . αg−1 j+1 gk+(r−1) αg−1 j+2 gk αg−1 j+2 gk+1 . . . αg−1 j+2 gk+(r−1) ... ... ... ... αg−1 j+r−1gk αg−1 j+r−1gk+1 . . . αg−1 j+r−1gk+(r−1)   , where l = {1, 2, 3, . . . , n 2 r2 } and where: φl : Hi 7→ Gr (hi)1 φl−→ g−1j gk (hi)2 φl−→ g−1j gk+1 ... ... ... (hi)r φl−→ g−1j gk+(r−1). . Here we notice that when when l = 1 then j = 1,k = 1, when l = 2 then j = 1,k = r + 1, when l = 3 then j = 1,k = 2r + 1, . . . when l = n r then j = 1,k = n − r + 1. When l = n r + 1 then j = r + 1,k = 1, when l = n r + 2 then j = r + 1,k = r + 1, when l = n r + 3 then j = r + 1,k = 2r + 1, . . . when l = 2n r then j = r + 1,k = n−r + 1, . . . , and so on. In [6], it is shown that the matrix Ω(v) can be written as: Ω(v) =   αg−111 g1 αg−112 g2 αg−113 g3 . . . αg−11n gn αg−121 g1 αg−122 g2 αg−123 g3 . . . αg−12n gn ... ... ... ... ... αg−1n1 g1 αg−1n2 g2 αg−1n3 g3 . . . αg−1nngn   , 96 A. Korban / J. Algebra Comb. Discrete Appl. 8(2) (2021) 91–105 where g−1ji are simply the elements of the group G. These elements are determined by how the matrix has been partitioned, what groups Hi of order r have been employed and how the maps φl have been defined to form the composite matrix. This representation of the composite matrix Ω(v) will make it easier to prove the upcoming results. For a given element v ∈ RG and some groups Hl of order r, we define the following code over the ring R : C(v) = 〈Ω(v)〉. (15) The code is formed by taking the row space of Ω(v) over the ring R. The code C(v) is a linear code over the ring R, since it is the row space of a generator matrix. It is not possible to determine the size of the code immediately from the matrix. In [6], it is shown that such codes are ideals in the group ring RG, and are held invariant by the action of the elements of G. Such codes are referred to as composite G-codes. We note that the matrix Ω(v) is an extension of the matrix σ(v) defined in [11]. Also, in [6], the authors show when the matrices Ω(v) are inequivalent to the matrices obtained from σ(v). This is one reason to study codes constructed from Ω(v)- this technique can produce codes which can not be obtained from codes constructed from σ(v) or other classical techniques. For example, please see [5] where many new binary self-dual codes are constructed via the composite matrices. 3. Composite G-codes and ideals in the group ring R∞G In this section, we show that the composite G- codes are ideals in the group ring R∞G and that the dual of the composite G- code is also a composite G- code in this setting. These two results are a simple generalization of Theorem 3.1 and Theorem 3.2 from [4]. We use the same arguments as in [4] to prove our results. For simplicity, we write each non-zero element in R∞ in the form γia where a = a0 + a1γ + · · ·+ · · · with a0 6= 0 and i ≥ 0, which means that a is a unit in R∞. We note that if v = γlg1 ag1g1 +γ lg2 ag2g2 +· · ·+γlgn agngn ∈ R∞G, then each row of Ω(v) corresponds to an element in R∞G of the following form: v∗j = n∑ i=1 γ lgji gi agjigigjigi, (16) where γlgji gi agjigi ∈ R∞, gi,gji ∈ G and j is the jth row of the matrix Ω(v). In other words, we can define the composite matrix Ω(v) as: Ω(v) =   γ lg11 g1 ag11g1 γ lg12 g2 ag12g2 γ lg13 g3 ag13g3 . . . γ lg1n gn ag1ngn γ lg21 g1 ag21g1 γ lg22 g2 ag22g2 γ lg23 g3 ag23g3 . . . γ lg2n gn ag2ngn ... ... ... ... ... γ lgn1 g1 agn1g1 γ lgn2 g2 agn2g2 γ lgn3 g3 agn3g3 . . . γ lgnn gn agnngn   , (17) where the elements gji are simply the group elements G. Which elements of G these are, depends how the composite matrix is defined, i.e., what groups we employ and how we define the φl map in individual blocks. Then we take the row space of the matrix Ω(v) over R∞ to get the corresponding composite G-code, namely C(v). 97 A. Korban / J. Algebra Comb. Discrete Appl. 8(2) (2021) 91–105 Theorem 3.1. Let R∞ be the formal power series ring and G a finite group of order n. Let Hi be finite groups of order r such that r is a factor of n with n > r and n,r 6= 1. Also, let v ∈ R∞G and let C(v) = 〈Ω(v)〉 be the corresponding code in Rn∞. Let I(v) be the set of elements of R∞G such that∑ γliaigi ∈ I(v) if and only if (γl1a1,γl2a2, . . . ,γlnan) ∈C(v). Then I(v) is a left ideal in R∞G. Proof. We saw above that the rows of Ω(v) consist precisely of the vectors that correspond to the elements of the form v∗j = ∑n i=1 γ lgji gi agjigigjigi in R∞G, where γ lgji gi agjigi ∈ R∞, gi,gji ∈ G and j is the jth row of the matrix Ω(v). Let a = ∑ γliaigi and b = ∑ γljbjgi be two elements in I(v), then a + b = ∑ (γliai + γ ljbj)gi, which corresponds to the sum of the corresponding elements in C(v). This implies that I(v) is closed under addition. Let w1 = ∑ γlibigi ∈ R∞G. Then if w2 corresponds to a vector in C(v), it is of the form ∑ (γljαj)v ∗ j . Then w1w2 = ∑ γlibigi ∑ (γljαj)v ∗ j = ∑ γlibiγ ljαjgiv ∗ j which corresponds to an element in C(v) and gives that the element is in I(v). Therefore I(v) is a left ideal of R∞G. Next we show that the dual of a composite G-code is also a composite G-code. Let I be an ideal in a group ring R∞G. Define R(C) = {w | vw = 0, ∀v ∈ I}. It follows that R(I) is an ideal of R∞G. Let v = γlg1 ag1g1 + γ lg2 ag2g2 + · · · + γlgn agngn ∈ R∞G and C(v) be the corresponding code. Let Ω : R∞G → Rn∞ be the canonical map that sends γlg1 ag1g1 + γlg2 ag2g2 + · · · + γlgn agngn to (γlg1 ag1,γ lg2 ag2, · · · ,γlgn agn ). Let I be the ideal Ω−1(C). Let w = (w1,w2, . . . ,wn) ∈ C⊥. Then the operator of product between any row of Ω(v) and w is zero: [(γ lgj1 g1 agj1g1,γ lgj2 g1 agj2g1, . . . ,γ lgjn g1 agjng1 ), (w1,w2, . . . ,wn)] = 0, ∀j. (18) Which gives n∑ i=1 γ lgji gi agjigiwi = 0, ∀j. (19) Let w = Ω−1(w) = ∑ γkgi wgigi and define w ∈ R∞G to be w = γkg1 bg1g1 + γkg2 bg2g2 + · · · + γkgn bgngn, where γkgi bgi = γ k g −1 i wg−1 i . (20) Then n∑ i=1 γ lgji gi agjigiwi = 0 =⇒ n∑ i=1 γ lgji gi agjigiγ k g −1 i bg−1 i = 0. (21) Here, gjigig −1 i = gji, thus this is the coefficient of gji in the product of w and v ∗ j , where v ∗ j is any row of the matrix Ω(v). This gives that w ∈R(I) if and only if w ∈C⊥. Let φ : Rn∞ → R∞G by φ(w) = w, then this map is a bijection between C⊥ and R(Ω−1(C)) = R(I). Theorem 3.2. Let C = C(v) be a code in R∞G formed from the vector v ∈ R∞G. Then Ω−1(C⊥) is an ideal of R∞G. Proof. The composite mapping Ω(φ(C⊥)) is permutation equivalent to C⊥ and φ(C⊥) is an ideal of R∞G. We know that φ is a bijection between C⊥ and R(Ω−1(C)), and we also know that Ω−1(C) is an ideal of R∞G as well. This proves that the dual of a composite G-code is also a composite G-code over the formal power series ring. 98 A. Korban / J. Algebra Comb. Discrete Appl. 8(2) (2021) 91–105 4. Projections and lifts of composite G-codes In this section, we extend more results from [4]. In fact, many of the results presented in this section are a consequence of the results proven in [8] and a simple generalization of the results proven in [4]. We first show that if v ∈ R∞G then Ω(v) is permutation equivalent to the matrix defined in Equa- tion 7. For simplicity, we write each non-zero element in R∞ in the form γia where a = a0 +a1γ+· · ·+· · · with a0 6= 0 and i ≥ 0, which means that a is a unit in R∞. Theorem 4.1. Let v = γlgi ag1g1 + γ lg2 ag2g2 + · · · + γlgn agngn ∈ R∞G, where agi are units in R∞. Let C be a finitely generated code over R∞. Then Ω(v) =   γ lg11 g1 ag11g1 γ lg12 g2 ag12g2 γ lg13 g3 ag13g3 . . . γ lg1n gn ag1ngn γ lg21 g1 ag21g1 γ lg22 g2 ag22g2 γ lg23 g3 ag23g3 . . . γ lg2n gn ag2ngn ... ... ... ... ... γ lgn1 g1 agn1g1 γ lgn2 g2 agn2g2 γ lgn3 g3 agn3g3 . . . γ lgnn gn agnngn   , is permutation equivalent to the standard generator matrix given in Equation 7. Proof. Take one non-zero element of the form γm0agi, where m0 is the minimal non-negative integer. By applying column and row permutations and by dividing a row by a unit, the element that corresponds to the first row and column of Ω(v) can be replaced by γm0. The elements in the first column of matrix Ω(v) have the form γlgj agj with lgj ≥ m0 and agj a unit, thus, these can be replaced by zero when they are added to the first row multiplied by −γlgj−m0 (agj )−1. Continuing the process using elementary operations, we obtain the standard generator matrix of the code C given in Equation 7. Example 4.2. Let G = 〈x,y | x4 = 1,y2 = x2,yxy−1 = x−1〉∼= Q8. Let v = ∑3 i=0 ( αi+1x i + αi+5x iy ) ∈ R∞Q8, where αi = αgi ∈ R∞. Let H1 = 〈a,b | a2 = b2 = 1,ab = ba〉 ∼= C2 × C2. We now define the composite matrix as: Ω(v) = ( A′1 A2 A3 A ′ 4 ) =   αg−11 g1 αg−11 g2 αg−11 g3 αg−11 g4 αg−11 g5 αg−11 g6 αg−11 g7 αg−11 g8 αφ1((h1)−12 (h1)1) αφ1((h1)−12 (h1)2) αφ1((h1)−12 (h1)3) αφ1((h1)−12 (h1)4) αg−12 g5 αg−12 g6 αg−12 g7 αg−12 g8 αφ1((h1)−13 (h1)1) αφ1((h1)−13 (h1)2) αφ1((h1)−13 (h1)3) αφ1((h1)−13 (h1)4) αg−13 g5 αg−13 g6 αg−13 g7 αg−13 g8 αφ1((h1)−14 (h1)1) αφ1((h1)−14 (h1)2) αφ1((h1)−14 (h1)3) αφ1((h1)−14 (h1)4) αg−14 g5 αg−14 g6 αg−14 g7 αg−14 g8 αg−15 g1 αg−15 g2 αg−15 g3 αg−15 g4 αg−15 g5 αg−15 g6 αg−15 g7 αg−15 g8 αg−16 g1 αg−16 g2 αg−16 g3 αg−16 g4 αφ4((h1)−12 (h1)1) αφ4((h1)−12 (h1)2) αφ4((h1)−12 (h1)3) αφ4((h1)−12 (h1)4) αg−17 g1 αg−17 g2 αg−17 g3 αg−17 g4 αφ4((h1)−13 (h1)1) αφ4((h1)−13 (h1)2) αφ4((h1)−13 (h1)3) αφ4((h1)−13 (h1)4) αg−18 g1 αg−18 g2 αg−18 g3 αg−18 g4 αφ4((h1)−14 (h1)1) αφ4((h1)−14 (h1)2) αφ4((h1)−14 (h1)3) αφ4((h1)−14 (h1)4)   , where: φ1 : (h1)i φ1−→ g−11 gi φ4 : (h1)i φ4−→ g−15 gj for i = {1,2,3,4} for when {i = 1, . . . ,4 and j = i + 4}, in A′1 and A ′ 4 respectively. This results in a composite matrix over R∞ of the following form: 99 A. Korban / J. Algebra Comb. Discrete Appl. 8(2) (2021) 91–105 Ω(v) =   X1 Y1 X2 Y1 X1 X3 X4 Y4 Y4 X4   =   α1 α2 α3 α4 α5 α6 α7 α8 α2 α1 α4 α3 α8 α5 α6 α7 α3 α4 α1 α2 α7 α8 α5 α6 α4 α3 α2 α1 α6 α7 α8 α5 α7 α6 α5 α8 α1 α4 α3 α2 α8 α7 α6 α5 α4 α1 α2 α3 α5 α8 α7 α6 α3 α2 α1 α4 α6 α5 α8 α7 α2 α3 α4 α1   . If we let v = γ2x3 + γ2(1 + γ)xy + γ2(1 + γ + γ2)x2y + γ2x3y ∈ R∞Q8, where 〈x,y〉∼= Q8, then C(v) = 〈Ω(v)〉 =   0 0 0 γ2 0 γ2(1 + γ) γ2(1 + γ + γ2) γ2 0 0 γ2 0 γ2 0 γ2(1 + γ) γ2(1 + γ + γ2) 0 γ2 0 0 γ2(1 + γ + γ2) γ2 0 γ2(1 + γ) γ2 0 0 0 γ2(1 + γ) γ2(1 + γ + γ2) γ2 0 γ2(1 + γ + γ2) γ2(1 + γ) 0 γ2 0 γ2 0 0 γ2 γ2(1 + γ + γ2) γ2(1 + γ) 0 γ2 0 0 0 0 γ2 γ2(1 + γ + γ2) γ2(1 + γ) 0 0 0 γ2 γ2(1 + γ) 0 γ2 γ2(1 + γ + γ2) 0 0 γ2 0   , and C(v) is equivalent to  γ2 0 0 0 0 γ2(1 + γ) γ2(1 + γ + γ2) γ2 0 γ2 0 0 γ2 0 γ2(1 + γ) γ2(1 + γ + γ2) 0 0 γ2 0 γ2(1 + γ + γ2) γ2 0 γ2(1 + γ) 0 0 0 γ2 γ2(1 + γ) γ2(1 + γ + γ2) γ2 0   . Clearly C(v) = 〈Ω(v)〉 is the [8, 4, 4] extended Hamming code. We now generalize the results from [4] on the projection of codes with a given type. Proposition 4.3. Let C be a composite G-code over R∞ of type {(γm0 )k0, (γm1 )k1, . . . , (γmr−1 )kr−1} with generator matrix Ω(v). The code generated by Ψi(Ω(v)) is a code over Ri of type {(γm0 )k0, (γm1 )k1, . . . , (γms−1 )ks−1} where ms is the largest mi that is less than e. Also, the code generated by Ψi(Ω(v)) is equal to {(Ψi(c1), Ψi(c2), . . . , Ψi(cn)) | (c1,c2, . . . ,cn) ∈C}. (22) Proof. If mi > e − 1 then Ψi sends γmiM′, where M′ is a matrix, to a zero matrix which gives the first part. The code C is formed by taking the row space of Ω(v) over the ring R∞, i.e. γl1a1v1 +γl2a2v2 +· · ·+ γlnanvn where γliai ∈ R∞ and vi are the rows of Ω(v). If w = γljajvj, then Ψi(w) = Ψi(γliai)Ψi(vi) by the equation given in (11) where Ψi(vi) applies the map coordinate-wise. This gives the second part. Since a composite G- code over R∞ is a linear code, the following results are a direct consequence of some results proven in [8]. We omit the proofs. Lemma 4.4. Let C be a composite G-code of length n over R∞, then, 100 A. Korban / J. Algebra Comb. Discrete Appl. 8(2) (2021) 91–105 (1) C⊥ has type 1m for some m, (2) C = (C⊥)⊥ if and only if C has type 1k for some k, (3) If C has a standard generator matrix G as in equation (7), then we have (i) the dual code C⊥ of C has a generator matrix H = ( B0,r B0,r−1 . . . B0,2 B0,1 Ikr ) , (23) where B0,j = − ∑j−1 l=1 B0,lA T r−j,r−l −A T r−j,r for all 1 ≤ j ≤ r; (ii) rank(C)+rank(C⊥)=n. Example 4.5. If we take the generator matrix G of a code C from Example 1, we can see that G =  γ2   1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1   γ2   0 1 + γ 1 + γ + γ2 1 1 0 1 + γ 1 + γ + γ2 1 + γ + γ2 1 0 1 + γ 1 + γ 1 + γ + γ2 1 0     , which is the standard generator matrix- here, A0,1 =   0 1 + γ 1 + γ + γ2 1 1 0 1 + γ 1 + γ + γ2 1 + γ + γ2 1 0 1 + γ 1 + γ 1 + γ + γ2 1 0   . In this case the generator matrix of the dual code C⊥ of C has the form: H = ( B0,1 Ik1 ) . Now, B0,1 = −AT0,1, thus H =   0 −(1 + γ) −(1 + γ + γ2) −1 1 0 0 0 −1 0 −(1 + γ) −(1 + γ + γ2) 0 1 0 0 −(1 + γ + γ2) −1 0 −(1 + γ) 0 0 1 0 −(1 + γ) −(1 + γ + γ2) −1 0 0 0 0 1   . We also have rank(C) + rank(C⊥) = 4 + 4 = 8 = n. Proposition 4.6. Let C be a self-orthogonal composite G-code over R∞. Then the code Ψi(C) is a self- orthogonal composite G-code over Ri for all i < ∞. Proof. We first show that Ψi(C) is self-orthogonal. Let v ∈ R∞G and 〈Ω(v)〉 = C(v) be the corre- sponding self-orthogonal composite G-code. This implies that [v,w] = 0 for all v,w ∈ 〈Ω(v)〉 = C(v). This gives that n∑ l=1 vlwl ≡ n∑ l=1 Ψi(vl)Ψi(wl)(mod γi) ≡ Ψi([v,w])(mod γi) ≡ 0 (mod γi). Hence Ψi(C) is a self-orthogonal code over Ri. To show that Ψi(C) is also a G-code, we notice that when taking Ψi(C) = Ψi(〈Ω(v)〉), it corresponds to Ψi(v) = Ψi(γlg1 ag1 )g1 + Ψi(γlg2 ag2 )g2 +· · ·+ Ψi(γlgn agn )gn, then Ψi(C) ∈ RiG. Thus Ψi(C) is also a composite G-code. 101 A. Korban / J. Algebra Comb. Discrete Appl. 8(2) (2021) 91–105 Definition 4.7. Let i,j be two integers such that 1 ≤ i ≤ j < ∞. We say that an [n,k] code C1 over Ri lifts to an [n,k] code C2 over Rj, denoted by C1 � C2, if C2 has a generator matrix G2 such that Ψ j i (G2) is a generator matrix of C1. We also denote C1 by Ψ j i (C2). If C is a [n,k] γ-adic code, then for any i < ∞, we call Ψi(C) a projection of C. We denote Ψi(C) by Ci. Lemma 4.8. Let C be a composite G-code over R∞ with type 1k. If Ω(v) is a standard form of C, then for any positive integer, i, Ψi(Ω(v)) is a standard form of Ψi(C). Proof. We know from Theorem 4.1 that Ω(v) is permutation equivalent to a standard form matrix defined in Equation 7. We also have that C has type 1k, hence Ψi(C) has type 1k. The rest of the proof is the same as in [8]. In the following, to avoid confusion, we let v∞ and v be elements of the group rings R∞G and RiG respectively. Let v∞ = γl1ag1g1 + γ l2ag2g2 + · · · + γlnagngn ∈ R∞G, and C(v∞) = 〈Ω(v∞)〉 be the corresponding composite G-code. Define the following map: Ω1 : R∞G →C(v∞), (γlg1 ag1g1 + γ lg2 ag2g2 + · · · + γ lgn agngn) 7→ M(R∞G,v∞). We define a projection of composite G-codes over R∞G to RiG. Let Ψi : R∞G → RiG (24) γia 7→ Ψ(γia). (25) The projection is a homomorphism which means that if I is an ideal of R∞G, then Ψi(I) is an ideal of RiG. We have the following commutative diagram: Rn∞G Ω1−→ C(v∞) Ψi ↓ ↓ Ψi Rni G −→ Ω1 C(v) . This gives that ΨiΩ1 = Ω1Ψi, which gives the following theorem. Theorem 4.9. If C is a composite G-code over R∞, then Ψi(C) is a composite G-code over Ri for all i < ∞. Proof. Let v∞ ∈ R∞G and C(v∞) be the corresponding composite G-code over R∞. Then Ω1(v∞) = C(v∞) is an ideal of R∞G. By the homomorphism in Equation 24 and the commutative diagram above, we know that Ψi(Ω1(v∞)) = Ω1(Ψi(v∞)) is an ideal of the group ring RiG. This implies that Ψi(C) is a composite G-code over Ri for all i < ∞. Theorem 4.10. Let C be a composite G-code over Ri, then the lift of C, C̃ over Rj, where j > i, is also a composite G-code. Proof. Let v1 = αg1g1 + αg2g2 + · · · + αgngn ∈ RiG and C = 〈Ω(v1)〉 be the corresponding composite G-code. Let v2 = βg1g1 + βg2g2 + · · · + βgngn ∈ RjG and C̃ = 〈Ω(v2)〉 be the corresponding composite G-code. We can say that v1 and v2 act as generators of C and C̃ respectively. We can clearly see that we can have Ψji (v2) = Ψ j i (βg1 )g1 + Ψ j i (βg2 )g2 + · · · + Ψ j i (βgn )gn = αg1g1 + αg2g2 + · · · + αgngn ∈ RiG, thus Ψji (v2) is a generator matrix of C. This implies that the composite G-code C(v1) over Ri lifts to a composite G-code over Rj, for all j > i. 102 A. Korban / J. Algebra Comb. Discrete Appl. 8(2) (2021) 91–105 The following results consider composite G-codes over chain rings that are projections of γ-adic codes. The results are just a simple consequence of the results proven in [8]. For details on notation and proofs, please refer to [8] and [4]. Lemma 4.11. Let C be a [n,k] composite G-code of type 1k, and G,H be a generator and parity-check matrices of C. Let Gi = Ψi(G) and Hi = Ψi(H). Then Gi and Hi are generator and parity check matrices of Ci respectively. Let i < j < ∞ be two positive integers, then (i) γj−iGi ≡ γj−iGj (mod γj); (ii) γj−iHi ≡ γj−iHj (mod γj). (iii) γj−1Ci ⊆Cj; (iv) v = γiv0 ∈Cj if and only if v0 ∈Cj−i; (v) Ker(Ψji)=γ iCj−i. Theorem 4.12. Let C be a composite G-code over R∞. Then the following two results hold. (i) the minimum Hamming distance dH(Ci) of Ci is equal to d = dH(C1) for all i < ∞; (ii) the minimum Hamming distance d∞ = dH(C) of C is at least d = dH(C1). The final two results we present in this section are a simple extension of the two results from [8] on MDS and MDR codes over R∞. We omit the proofs since a composite G- code over R∞ is a linear code and for that fact, the proofs are the same as in [8]. Theorem 4.13. Let C be a composite G-code over R∞. If C is an MDR or MDS code then C⊥ is an MDS code. Theorem 4.14. Let C be a composite G-code over Ri, and C̃ be a lift of C over Rj, where j > i. If C is an MDS code over Ri then the code C̃ is an MDS code over Rj. 5. Self-dual γ-adic composite G-codes In this section, we extend some results for self-dual γ-adic codes to composite G-codes over R∞. As in previous sections, the results presented here are just a simple generalization of the results proven in [8] and [4]. Fix the ring R∞ with R∞ →···→ Ri →···→ R2 → R1 and R1 = Fq where q = pr for some prime p and nonnegative integer r. The field Fq is said to be the underlying field of the rings. We now generalize four theorems from [8]. The first two consider self-dual codes over Ri with a specific type and projections of self-dual codes over R∞ respectively. The third one considers a method for constructing self-dual codes over F from a self-dual code over Ri. We extend these to self-dual composite G-codes over Ri and R∞ respectively. Theorem 5.1. Let i be odd and C be a composite G-code over Ri with type 1k0 (γ)k1 (γ2)k2 . . . (γi−1)ki−1. Then C is a self-dual code if and only if C is self-orthogonal and kj = ki−j for all j. Proof. It is enough to show that Ω(v) where v ∈ RiG and G is a finite group, is permutation equivalent to the matrix (3). The rest of the proof is the same as in [8]. 103 A. Korban / J. Algebra Comb. Discrete Appl. 8(2) (2021) 91–105 Theorem 5.2. If C is a self-dual composite G-code of length n over R∞ then Ψi(C) is a self-dual composite G-code of length n over Ri for all i < ∞. Proof. This is a direct consequence of Theorem 3.4 in [8] and Proposition 4.4 of this work. Theorem 5.3. Let i be odd. A self-dual composite G-code of length n over Ri induces a self-dual composite G-code of length n over Fq. Proof. The first part of the proof is identical to the one of Theorem 5.5 from [4]. Secondly, when the map Ψi1(G̃) is used in [8], we notice that in our case the map will correspond to Ψ i 1(G̃) = Ψ i 1(v) = Ψi1(γ lg1 ag1 )g1 + Ψ i 1(γ lg2 ag2 )g2 + · · · + Ψi1(γlgn agn )gn, assuming that G̃ is the generator matrix of a com- posite G-code and v ∈ RiG. Then Ψi1(G̃) is the generator matrix of a composite G-code over Fq. Theorem 5.4. Let R = Re be a finite chain ring, F = R/〈γ〉, where |F| = q = pr, 2 6= p is a prime. Then any self-dual composite G-code C over F can be lifted to a self-dual composite G-code over R∞. Proof. From Theorem 4.10 we know that a composite G-code over Ri can be lifted to a composite G- code over Rj, where j > i. To show that a self-dual composite G-code over F lifts to a self-dual composite G-code over R∞, it is enough to follow the proof in [8]. 6. Composite G-codes over principal ideal rings In this section, we study composite G-codes over principal ideal rings. We study codes over this class of rings by the generalized Chinese Remainder Theorem. Please see [2] for more details on the notation and definitions of the principal ideal rings. Let R1e1,R 2 e2 , . . . ,Rses be chain rings, where R j ej has unique maximal ideal 〈γj〉 and the nilpotency index of γj is ej. Let Fj = Rjej/〈γj〉. Let A = CRT(R1e1, . . . ,R j ej , . . . ,Rses ). We know that A is a principal ideal ring. For any 1 ≤ i < ∞, let A j i = CRT(R 1 e1 , . . . ,R j i , . . . ,R s es ). This gives that all the rings Aji are principal ideal rings. In particular, A j ej = A. We denote CRT(R1e1 . . . ,R j ∞, . . . ,R s es ) by Aj∞. For 1 ≤ i < ∞, let Cji be a code over R j i . Let Cji = CRT(C 1 e1 , . . . ,Cji , . . . ,C s es ) be the associated code over Aji. Let Cj∞ = CRT(C 1 e1 , . . . ,Cj∞, . . . ,C s es ) be associated code over Aj∞. We can now prove the following. Theorem 6.1. Let Cjej be a composite G-code over the chain ring R j ej that is Cjej is an ideal in RejG. Then Cj∞ =CRT(C1e1, . . . , C j ∞, . . . ,Cses ) is a composite G-code over A j ∞. Proof. Let vj ∈Cjej. We know that v ∗ j also belongs to C j ej where v∗j has the form defined in (16). Let v ∈ Cj∞. Now if v = CRT(v1,v2, . . . ,vs), then v∗ = CRT(v∗1,v∗2, . . . ,v∗s) and so v∗ ∈ Cj∞ giving that Cj∞ is an ideal in Aj∞G, and thus giving that Cj∞ is a composite G-code over Aj∞. 104 A. Korban / J. Algebra Comb. Discrete Appl. 8(2) (2021) 91–105 7. Conclusion In this work, we generalized the known results on G-codes over the formal power series rings and finite chain rings Fq[t]/(ti) to composite G-codes over the same alphabets. We showed that the dual of a composite G-code is also a composite G-code and we studied the projections and lifts of the composite G-codes with a given type in this setting. We extended many theoretical results on γ-adic G-codes and G-codes over principal ideal rings to composite γ-adic G-codes and composite G-codes over principal ideal rings. Since the results presented in this paper and in [4] only consider the finite chain rings Fq[t]/(ti), it is suggested that for future research, these families of codes; G - Codes and composite G - Codes, are studied over a more general finite chain rings as it was done using a unified treatment in [1]. References [1] R. L. Bouzara, K. Guenda, E. Martinez-Moro, Lifted codes and lattices from codes over finite chain rings, arXiv:2007.05871. [2] S. T. Dougherty, Algebraic coding theory over finite commutative rings, SpringerBriefs in Mathe- matics Springer (2017). [3] S. T. Dougherty, J. Gildea, A. Korban, Extending an established isomorphism between group rings and a subring of the n×n matrices, International Journal of Algebra and Computation, Published: 25 February 2021. [4] S. T. Dougherty, J. Gildea, A. Korban, G- codes over formal power series rings and finite chain rings, J. Algebra Comb. Discrete Appl. 7 (2020) 55–71. [5] S. T. Dougherty, J. Gildea, A. Korban, A. 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