Independence of Vectors in Codes over Rings Steven T. Dougherty Department of Mathematics University of Scranton Scranton, PA 18510, USA Email: [email protected] and Hongwei Liu ∗ Department of Mathematics Huazhong Normal University Wuhan, Hubei 430079, China Email: h w [email protected] June 22, 2011

Abstract We study codes over Frobenius rings. We describe Frobenius rings via an isomorphism to the product of local Frobenius rings and use this decomposition to describe an analog of linear independence. Special attention is given to codes over principal ideal rings and a basis for codes over principal ideal rings is defined. We prove that a basis exists for any code over a principal ideal ring and that any two basis have the same number of vectors.

Key Words: Frobenius rings, principal ideal rings, codes over rings, Chinese Remainder Theorem. MSC: Primary 94B05, Secondary 13A99.

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The second author is supported by the National Natural Science Foundation of China (10571067)

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1

Introduction

Coding theory began with the study of codes as subsets of binary vector spaces. It expanded to consider codes over the fields F3 and F4 and later, especially with respect to designs, codes over Fp were studied. More recently, mathematicians’ interest in codes over finite rings was aroused again by the paper written by Hammons et al. (see [5]). In this paper, it is proved that some interesting non-linear codes, such as the Kerdock, Preparata, and Goethal codes can be viewed as linear codes over Z4 via the Gray map from Zn4 to F2n 2 , and that their n apparent duality could be described in terms of their duality in Z4 . In recent years, many papers on codes over finite rings, such as the ring Zm , finite chain rings, and Galois rings, have appeared. In Park’s paper [12] he describes the independence of vectors over Zm . We extend his results considering codes over Frobenius rings. Many well-known finite rings are Frobenius rings, for example, Zm , finite chain rings, and principal ideal rings are Frobenius. We shall study some of the foundational results of coding theory with respect to codes over Frobenius rings. The paper is organized as follows. In Section 2, we give some notations and some basic results for codes over Frobenius rings. In Section 3, we will focus on codes over finite Frobenius rings and we shall describe the representation of elements in a local ring. In Section 4, we study codes over a special class of Frobenius rings, namely principal ideal rings.

2

Some Notations and Basic Results of Codes over Finite Rings

Throughout the work we shall assume that all rings are commutative, finite and have a multiplicative unity. We shall define some types of rings that we shall consider in the paper. A finite commutative ring R is Frobenius if the R-module R is injective. Alternatively, we can say a finite commutative ring is Frobenius if R/J(R) is isomorphic to soc(R) where J(R) is the Jacobson radical of the ring R and soc(R) is the socle of the ring. Recall that the Jacobson radical is the intersection of all maximal ideals in the ring and the socle of the ring is the sum of the minimal R-submodules. A ring is a local ring if it has a unique maximal ideal. A principal ideal ring is a ring such that every ideal is generated by a single element. Let R be a finite commutative ring. There exist ideals a1 , a2 , · · · , as of R, relatively prime T in pairs, such that sj=1 aj = {0}, see ([10], p. 95). There is a canonical homomorphism from R to R/aj , namely Φj : R → R/aj , with Φj (r) = r + aj . 2

It is extended naturally to the following: Φ : R → R/a1 × R/a2 × · · · × R/as .

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It is easy to see that the map above is an isomorphism. We let Rj = R/aj and then extend the map coordinatewise to s Y Φ : Rn → Rjn . (2) j=1

By the Chinese Remainder Theorem the inverse map is an isomorphism. We denote the inverse of this map by CRT. For Frobenius rings we can say more. Namely, we have the following theorem which can be found in ([10], p. 224). Theorem 2.1. Let R be a Frobenius ring, then R∼ = CRT(R1 , R2 , · · · , Rs ), where Ri is a local Frobenius ring. That is Rj = R/aj is a local Frobenius ring for each j. We shall use this decomposition of rings to understand codes defined over Frobenius rings. A code C of length n over a ring R is a subset of Rn . We say that a code C is linear if it is an R-submodule of Rn . The Hamming distance between two vectors is the number of coordinates in which they disagree. The Hamming weight of a vector is the number of non-zero coordinates of that vector. For a linear code, the minimum Hamming distance of the code is equal to the minimum Hamming weight. We let denote an involution of R and we attach the standard inner-product to the space n R . Specifically, X [v, w] = vi w i . (3) The involution can be the identity but we define it this way to be as general as possible. For a code C, its dual code is defined as follows: C ⊥ = {u | [u, c] = 0, ∀ c ∈ C}.

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It is well known that for codes over Frobenius rings, |C||C ⊥ | = |R|n , (see [13] for a proof). Let Ci be a code over Ri , where Ri is defined in Equation (3.1), of length n and let C = CRT(C1 , C2 , · · · , Cs ) = Φ−1 (C1 × C2 × · · · × Cs ) = {Φ−1 (c1 , c2 , · · · , cs )|cj ∈ Cj }.

We call C the Chinese product of codes C1 , C2 , · · · , Cs . 3

Suppose, as defined above, Φ : R → R1 × R2 × · · · × Rs

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r 7→ (r1 , r2 , · · · , rs ).

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with We denote Φi : R → Ri with r 7→ ri . If R is Frobenius then each Ri is a local Frobenius ring with mi the unique maximal ideal. If a is not a unit in Ri , then hai 6= Ri . The ideal hai is contained in a maximal ideal, so hai ⊆ mi , that means a ∈ mi , so mi contains all non-units of Ri . This gives the following well-known lemma. Lemma 2.2. Let Ri be a finite local ring and let mi be the unique maximal ideal, then mi contains all non-units of Ri . Let C be a linear code of length n over R, then Φi (C) is a linear code over Ri since Φi is a homomorphism. This allows us to reduce the study of codes over finite Frobenius rings to that of codes over finite local Frobenius rings. Remark 1. The following is an example of a ring that is a local Frobenius ring but not a chain ring. We shall use this ring to exhibit several of the results of the paper. Example 1. Let R = Z4 [x]/hx2 i, where hx2 i denotes the ideal generated by x2 . Recall that the Jacobson radical of ring R is denoted by J(R). Then J(R) = h2, xi, the ideal generated by 2 and x, and R/J(R) = Z2 , so R is a local Frobenius ring. But J(R) cannot be generated by one element, hence R is not a chain ring. In fact, J(R)2 = h2xi, J(R)3 = {0}, so the ideal h2i and the ideal hxi are not a power of the radical. For a linear code C of length n over R, we define rC = min{l | there exists a monomorphism C → Rl as R − modules}. We say that a linear code C over R is free if C is isomorphic as a module to Rt for some t. It is immediate that if C is free then rC = t, where C ∼ = Rt . Lemma 2.3. Assume the notations given above. If C is free, then Ci is free for all i.

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Proof. If Ci is not free for some i, then there exists v ∈ Ci with |hvi| < |Ri |, and v 6= αw, w ∈ Ci , where α is a non-unit. Note that any free code contains v = αw, where α is a non-unit and |hvi| < |Ri |, but the key is that w is not in the code Ci . But v = Φi (x), x ∈ C, then |hxi| < |R|. If |hxi| = |R|, then there exists at least one xj such that xj is a unit in R, where x = (x1 , · · · , xj , · · · , xn ), then since Φi is a homomorphism, Φi (xj ) is a unit in Ri , and |hvi| = |hΦi (x)i| = |h(Φi (x1 ), · · · , Φi (xj ), · · · , Φi (xn ))i| = |Ri |.

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This is a contradiction, so R is not free. Following these concepts and lemmas above, we have the following theorem. Theorem 2.4. Let C1 , C2 , · · · , Cs be codes of length n, with Ci a code over Ri for i = 1, 2, · · · , s, and let C = CRT(C1 , C2 , · · · Cs ). Then (i) |C| = Πsi=1 |Ci |; (ii) rC = max{rCi |1 ≤ i ≤ s}; (iii) C is a free code if and only if each Ci is a free code with the same rCi . Proof. (i) This identity can be obtained immediately from the following isomorphism C = CRT(C1 , C2 , · · · Cs ) ∼ = C1 × C2 × · · · × Cs .

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(ii) Let rCi = li , and max{rCi |1 ≤ i ≤ s} = l. Suppose ψ : Ci → Rili is the monomorphism from Ci to Rili such that rCi = li . Then the composition homomorphism µi : Ci → Rili ,→ Ril is also a monomorphism from Ci to Ril , where the first homomorphism is ψ and the second homomorphism Rili ,→ Ril is the usual embedding homomorphism. This can be extended to a monomorphism as follows. Let (µ1 , · · · , µs ) : C1 × · · · × Cs → R1l × · · · × Rsl .

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Since C = CRT(C1 , · · · , Cs ), and R1l × · · · × Rsl ∼ = (R1 × · · · × Rs )l ∼ = Rl , we have a monomorphism induced by (µ1 , · · · , µs ), denoted by µ, µ : C = CRT(C1 , · · · , Cs ) → Rl . According to the definition of µi , l is also the minimal number such that a mapping C → Rl is a monomorphism, and this implies that rC = l. So the result in (ii) holds. (iii) Let Ci be free codes such that all of them have the same rCi . Then we have that rCi = r, where Ci ∼ = Rir . By (i) we have, |C| = Πsi=1 |Ci | = Πsi=1 |Ri |r = (Πsi=1 |Ri |)r = |R|r , 5

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and by (ii), rC = max{rCi |1 ≤ i ≤ s} = r. This implies that C → RrC is an isomorphism, that is C ∼ = Rr . So we get that C is free. If C is free with rC = r, and C ∼ = Rr , then |C| = |R|r , and all Ci must be free by Lemma 2.3. If there exist at least two rCi which are not equal, suppose Cj is the code whose rCj is the largest, that is rCj = r by (ii), then there exists at least one rCi < r, j 6= i. So we get |C| = Πsi=1 |Ci | = Πsi=1 |Ri |rCi < (Πsi=1 |Ri |)r = |R|r , which is a contradiction. Let dH (C) denote the minimal Hamming distance of code C. We have the following well known lemma. We include a short proof for completeness. Lemma 2.5. Let C1 , C2 , · · · , Cs be codes with Ci a code over Ri . Then dH (CRT(C1 , C2 , · · · , Cs )) = min{dH (Ci )}. Proof. We have dH (CRT(C1 , C2 , · · · , Cs )) = min{dH (Φ−1 (c1 , c2 , · · · , cs ))|ci ∈ Ci )} = min{dH (Φ−1 (0, · · · , 0, ci , 0, · · · , 0))|ci ∈ Ci } = min{dH (Ci )}. It is well known (see [8] for example) that for codes C of length n over any alphabet of size m dH (C) ≤ n − logm (|C|) + 1. (11) Codes meeting this bound are called MDS (M aximal Distance Separable) codes. Furthermore, if C is linear (see [6]), then dH (C) ≤ n − rC + 1.

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Codes meeting this bound are called MDR (M aximal Distance with respect to Rank) codes. Notice that Rn is a MDR code for any R as a trivial example but so is any ideal of R, since for an ideal m of R, rm = 1 and the minimum distance is 1. However R is the only MDS code of length 1. Theorem 2.6. Let C1 , C2 , · · · , Cs be codes over Ri . If Ci is an MDR code for each i, then C = CRT(C1 , C2 , · · · , Cs ) is an MDR code. If Ci is an MDS code with the same rCi for each i, then C = CRT(C1 , C2 , · · · , Cs ) is an MDS code. Proof. Let rCi be the minimal number such that there is a monomorphism from rCi Ci → Ri . By Theorem 2.4, Lemma 2.5 and the bound in (12), we have dH (CRT(C1 , C2 , · · · , Cs )) = min{dH (Ci )} = min{n − rCi + 1 | 1 ≤ i ≤ s} = n − max{rCi | 1 ≤ i ≤ s} + 1 = n − rC + 1. 6

By Theorem 2.4, we have that if each code Ci is MDS, then each is free (and they have the same ranks). An MDS code must be free to meet the Singleton bound, i.e. it must have cardinality |R|k where k is its rank. Therefore CRT(C1 , C2 , · · · , Cs ) is also free and hence MDS. To each ring Rj we attach an involution, possibly the identity such that for a ∈ Rj , Φj (a) = Φj (a) where a is the involution of R. For a code C of length n over R, using the inner-product in (3), we have the following theorem. Theorem 2.7. Assume that CRT(a1 , . . . , as ) = CRT(a1 , . . . , as ). If C = CRT(C1 , C2 , · · · , Cs ), then C ⊥ = CRT(C1⊥ , C2⊥ , · · · , Cs⊥ ). Proof. Let u, v ∈ Rn . We first show that [u, v] = 0 if and only if [Φj (u), Φj (v)] = 0, ∀j = 1, 2, · · · , s.

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It is easy to obtain that [u, v] = 0 if and only if [Φ(u), Φ(v)] = 0, since Φ is an isomorphism. Namely, given that CRT(a1 , . . . , as ) = CRT(a1 , . . . , as ), we have that [Φ(u), Φ(v)] = ([Φ1 (u), Φ1 (v)], [Φ2 (u), Φ2 (v)], · · · , [Φs (u), Φs (v)]) = Φ([u, v]), viewing [Φ(u), Φ(v)] as an element of R1 × R2 × · · · × Rs . According to the definition of C ⊥ , w ∈ C ⊥ ⇔ [w, c] = 0, ∀c ∈ C. By equation (13), we have [w, c] = 0 if and only if [Φj (w), Φj (c)] = 0, ∀j = 1, 2, · · · , s. This implies that w ∈ C ⊥ ⇔ w ∈ CRT(C1⊥ , C2⊥ , · · · , Cs⊥ ).

3 3.1

Codes over Frobenius Rings Independence of vectors over local Frobenius rings

Let R be a finite Frobenius ring and let Ri be a local Frobenius ring. We begin with some definitions and lemmas over finite local Frobenius rings. Definition 1. Let Ri be a local Frobenius ring with unique maximal ideal mi , and let w1 , · · · , ws be vectors in Rin . Then w1 , · · · , ws are modular independent if and only if P αj wj = 0 implies that αj ∈ mi for all j. 7

Notice that fields are local Frobenius rings, where their maximal ideal is {0}. The definition of modular independent above is a generalization of linear independence over fields. The zero vector, denoted 0, is modular dependent, and any nonzero vector w in Rin is modular independent. It is easy to obtain that if w1 , · · · , ws are modular independent, then wj 6= 0 for all j. Let w1 , · · · , ws be vectors in Rin . As usual, we denote the set of all linear combinations of w1 , · · · , ws by hw1 , · · · , ws i. We have the following lemma. Lemma 3.1. Let w1 , · · · , ws ∈ Rin . Then w1 , · · · , ws are modular dependent if and only if some wj can be written as a linear combination of the other vectors. Proof. Suppose w1 , · · · , ws are modular dependent. Then there exists αj ∈ Ri with P at least one αj 6∈ mi , such that αj wj = 0. By Lemma 2.2, mi contains all the non-unit elements, so αj is a unit. This gives wj = (−αj−1 α1 )w1 + · · · + (−αj−1 αj−1 )wj−1 + (−αj−1 αj+1 )wj+1 + · · · + (−αj−1 αs )ws . Without loss of generality, suppose ws can be written as a linear combination of the other vectors. Then s−1 X ws = αj wj . j=1

This gives α1 w1 + · · · + αs−1 ws−1 + (−1)ws = 0 as −1 is a unit in Ri and this implies that w1 , · · · , ws are modular dependent. By Theorem 2.1 and the definitions of Φ and Φi , we can use the definition of modular independence of vectors in Rin to define modular independence in Rn as follows. Definition 2. The vectors v1 , · · · , vk in Rn are modular independent if Φi (v1 ), · · · , Φi (vk ) are modular independent for some i. P Theorem 3.2. If v1 , · · · , vk are modular independent over R and αj vj = 0, then αj is not a unit for all j. Proof. Let i be the index such that Φi (v1 ), · · · , Φi (vk ) are modular independent. Then P αj vj = 0 implies that Φi (αj )Φi (vj ) = 0, and Φi (αj ) ∈ mi is not a unit for all j. If αj were a unit in R then there exists β ∈ R with αj β = 1, then Φi (αj )Φi (β) = 1, and Φi (αj ) is a unit, this is a contradiction. The converse is not true. For example, (2, 2) and (3, 3) over Z6 have the properties that if α1 (2, 2) + α2 (3, 3) = (0, 0) then αi must be non-units but they are not modular independent P

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over Z6 , since their images under Φ2 and Φ3 are not modular independent over Z2 and Z3 respectively. This leads to the following definition. Definition 3. Let v1 , · · · , vk be non-zero vectors in Rn . Then v1 , · · · , vk are independent P if αj vj = 0 implies that αj vj = 0 for all j. Theorem 3.3. If v1 , · · · , vk are independent and αw 6∈ hv1 , · · · , vk i, for any α 6= 0, then v1 , · · · , vk , w are independent. P P Proof. If αj vj + βw = 0, then αj vj = −βw, which is a contradiction unless P β = 0. In which case, αj vj = 0, and then αj vj = 0 for all j since v1 , · · · , vk are independent. Following Definition 3, we can easily get the following lemma. Lemma 3.4. If the nonzero vectors w1 , · · · , ws in Rin are independent, then they are modular independent over Ri . P Proof. Suppose αj wj = 0. Then αj wj = 0 for all j. If αj 6∈ mi for some j, then αj is a unit, and this implies that wj = 0. Central to the study of algebraic coding theory is the concept of a code’s generator matrix. The rows of the generator matrix form a basis of the code. We shall now give a definition of a basis of a code over a finite ring. Definition 4. Let C be a code over R. The codewords c1 , c2 , · · · , ck are called a basis of C if they are independent, modular independent and generate C. Remark 2. We note that modular independence does not imply independence nor does independence imply modular independence. For example, let (11, 7) and (3, 9) be vectors over Z12 . Since (11, 7) and (3, 9) map to (3, 3) and (3, 1) over Z4 and to (2, 1) and (0, 0) over Z3 via the natural homomorphisms, they are modular independent by definition. However, we have that 6(11, 7)+2(3, 9) = (0, 0), and 6(11, 7) = (6, 6) = 2(3, 9) 6= (0, 0). This gives that (11, 7), (3, 9) are not independent over Z12 . This shows that modular independence does not imply independence. Let (4, 0) and (0, 3) be vectors over Z12 . If a(4, 0) + b(0, 3) = (0, 0), then 4a ≡ 0 and 3b ≡ 0, which implies that a = 0, 3, 6, 9 and b = 0, 4, 8, so a(4, 0) = (0, 0) and b(0, 3) = (0, 0). This gives that (4, 0) and (0, 3) are independent. But (4, 0) and (0, 3) map to (0, 0) and (0, 3) over Z4 and map to (1, 0) and (0, 0) over Z3 via the natural homomorphisms. Since 1(0, 0) + 0(0, 3) = (0, 0) and 0(1, 0) + 1(0, 0) = (0, 0), we get that (4, 0) and (0, 3) are not modular independent over Z12 . This shows that independence does not imply modular independence. This remark shows why we require both independence and modular independence for a basis. In the next subsection we show that any code over a principal ideal ring has such a basis and that the number of elements in any two bases are the same. 9

3.2

Representation of elements in local rings

In this subsection we shall show that elements in a local Frobenius ring can be represented in a way similar to those in a chain ring. Let R be a finite local Frobenius ring with unique maximal ideal m. We know that F = R/m is a field. Assume that the characteristic of the field is p with |F | = ps . Define µ : R → F, r 7→ r + m to be the natural homomorphism from R to its residue field F . Then µ can be extended naturally from Rn to F n by µ : Rn → F n , u = (a0 , a1 , · · · , an−1 ) 7→ (a0 + m, a1 + m, · · · , an−1 + m). We have the following chain of ideals: R = m0 ⊇ m ⊇ m2 ⊇ · · · ⊇ me = {0}.

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The number e is the minimal such that me = {0}. This number is the nilpotency index of m. See ([10], p. 84) for a proof of this fact. There can be ideals not in this chain but they are still contained in the unique maximal ideal. Example 2. Following Example 1, let R = Z4 [x]/hx2 i, where hx2 i denotes the ideal generated by x2 . The maximal ideal of R is m = h2, xi, and we have R = Z4 [x]/hx2 i ⊇ m ⊇ m2 ⊇ m3 = {0}. For each i = 0, 1, . . . , e − 1, we fix a choice of elements ci0 , ci1 , . . . , cisi of R such that mi /mi+1 = {ci0 + mi+1 , ci1 + mi+1 , . . . , cisi + mi+1 }. That is, we fix a unique representative for each coset in mi /mi+1 . Of course, there are other possible labelings but we fix this particular labeling. Proposition 3.5. Let R be a finite local Frobenius ring with maximal ideal m. Assume the labeling of the cosets in mi /mi+1 given above, then for any r ∈ R, there exist unique P α0 , α1 , · · · , αe−1 such that αi ∈ {ci0 , ci1 , . . . , cisi }, i = 0, 1, . . . , e − 1, and r = e−1 i=0 αi . Proof. We prove the proposition by finite induction. Let r ∈ R, then r is in a unique coset of m. Suppose r ∈ α0 + m, then r = α0 + τ1 , α0 + m ∈ R/m, τ1 ∈ m. Suppose, for j < e − 1, we have r = α0 + α1 + · · · + αj−1 + τj , αi + mi ∈ mi−1 /mi , τj ∈ mj , 10

then τj is in a coset of mj+1 . Suppose τj ∈ αj + mj+1 , this implies that τj = αj + τj+1 , αj + mj+1 ∈ mj /mj+1 , τj+1 ∈ mj+1 . That is r = α0 + α1 + · · · + αj−1 + αj + τj+1 , where αi + mi+1 ∈ mi /mi+1 , τj+1 ∈ mj+1 for i = 1, · · · , j. Then this implies that r = α0 + α1 + · · · + αe−2 + αe−1 + τe , τe ∈ me = {0}. We get r = α0 + α1 + · · · + αe−2 + αe−1 , αi + mi+1 ∈ mi /mi+1 . The fact that |R|/|m| · |m|/m2 | · |m2 |/m3 | · · · |me−1 |/me | = |R|/1 = |R|, gives that the representation is unique. Example 3. Now we can take Example 1 to explain how the elements of R correspond to the sum of the elements in mi /mi+1 . Notice that R = Z4 [x]/hx2 i = {a+bx+hx2 i | a, b ∈ Z4 }, and m = h2, xi = h2i + hxi = {2a + bx | a, b ∈ Z4 }. The 16 elements in R are as follows: {0, 1, 2, 3, x, 2x, 3x, 1 + x, 2 + x, 3 + x, 1 + 2x, 2 + 2x, 3 + 2x, 1 + 3x, 2 + 3x, 3 + 3x}. It is well-known that there is a relation in R with α ∼ β ⇔ α − β ∈ m, then the elements of R are divided into two classes {0, 2, x, 2 + x, 2x, 2 + 2x, 3x, 2 + 3x} and {1, 3, 1 + x, 1 + 2x, 1 + 3x, 3 + x, 3 + 2x, 3 + 3x}. Notice that m2 = {0, 2x} and 2 + m2 = {2, 2 + 2x}, x + m2 = {x, 3x}, (2 + x) + m2 = {2 + x, 2 + 3x}. Then we have R/m = {0 = m, 1 = 1 + m}, m/m2 = {m2 , 2 + m2 , x + m2 , (2 + x) + m2 }, and m2 /m3 = {{0}, {2x}} = {0 + m3 , 2x + m3 }. Then the decompositions are: 0 2 x 3x 2+x 1 + 2x 3 + 2x 2 + 3x

= = = = = = = =

0 + 0 + 0, 0 + 2 + 0, 0 + x + 0, 0 + x + 2x, 0 + (2 + x) + 0, 1 + 0 + 2x, 1 + 2 + 2x, 0 + (2 + x) + 2x,

1 = 1 + 0 + 0, 3 = 1 + 2 + 0, 2x = 0 + 0 + 2x, 1 + x = 1 + x + 0, 3 + x = 1 + (2 + x) + 0, 2 + 2x = 0 + 2 + 2x, 1 + 3x = 1 + x + 2x, 3 + 3x = 1 + (2 + x) + 2x. 11

4

Codes over Principal Ideal Rings

In this section, we concentrate our study on codes over principal ideal rings. We generalize the results from the paper by Park [12] for codes over Zm to codes over principal ideal rings. We know that principal ideal rings are Frobenius rings (see Proposition 2.9 in [1]). Let R be a principal ideal ring. From Proposition 2.8 of [1], there exists an isomorphism Φ : R → R1 × · · · × Rs ,

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where Ri is a chain ring for all i. That is R = CRT(R1 , R2 , · · · , Rs ). For r ∈ R, write Φ(r) = (r1 , · · · , rs ), where ri ∈ Ri . Then we have the following homomorphism, denoted by Φi , Φi : R → Ri , r 7→ ri . Let γi be a generator of the unique maximal ideal of Ri with nilpotency index ei . Suppose βi is the inverse image of (1, · · · , 1, γi , 1, · · · , 1) under the isomorphism Φ in (15), that is Φ(βi ) = (1, · · · , 1, γi , 1, · · · , 1). Lemma 4.1. Assume the notations given above. Then Ker(Φi ) = hβiei i. Proof. If r ∈ hβiei i, then r = ββiei for some β ∈ R, and this gives that Φi (r) = Φi (ββiei ) = Φi (β)Φi (βiei ) = Φi (β)Φi (Φ−1 ((1, · · · , 1, γi , 1, · · · , 1)ei )) = Φi (β)0 = 0. This implies that hβiei i ⊆ Ker(Φi ). Suppose a ∈ Ker(Φi ), and Φ(a) = (a1 , · · · , ai−1 , ai , ai+1 , · · · , as ), then ai = 0. We have Φ(a) = = = =

(a1 , · · · , ai−1 , 0, ai+1 , · · · , as ) = (a1 , · · · , ai−1 , γiei , ai+1 , · · · , as ) (a1 , · · · , ai−1 , 1, ai+1 , · · · , as )(1, · · · , 1, γiei , 1, · · · , 1) (a1 , · · · , ai−1 , 1, ai+1 , · · · , as )(1, · · · , 1, γi , 1, · · · , 1)ei Φ(λ)Φ(βiei ) = Φ(λβiei ),

where λ ∈ R and Φ(λ) = (a1 , · · · , ai−1 , 1, ai+1 , · · · , as ). Then a = λβiei ∈ hβiei i, since Φ is an isomorphism. This implies that Ker(Φi ) ⊆ hβiei i. We give an example to explain the lemma above. Example 4. Let Φ : Z36 → Z4 × Z9 be the isomorphism under the Chinese Remainder Theorem. Let Φ4 : Z36 → Z4 and Φ9 : Z36 → Z9 be the respective homomorphisms. We know that the unique maximal ideal in Z4 is generated by 2, and the unique maximal ideal

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is generated by 3 in Z9 respectively, and the nilpotency indexes of 2 and 3 are both 2. The inverse images of (2, 1) and (1, 3) in Z4 × Z9 are 10 and 21 in Z36 . We have h102 i = h28i = {0, 4, 8, 12, 16, 20, 24, 28, 32} = Ker(Φ4 ), and h212 i = h9i = {0, 9, 18, 27} = Ker(Φ9 ). In order to prove that any two bases of a code over a finite principal ideal ring have the same number of vectors, we shall first prepare some lemmas and theorems. The following lemma can be found in [11]. Lemma 4.2. Assume the notations given above. For any 0 = 6 ri ∈ Ri , there is a unique j, j 0 ≤ j < ei such that ri = ui γi with ui a unit. The unit ui is unique modulo γiei −j only. Lemma 4.3. Assume the notations given above. Let R be a principal ideal ring, then for any 0 6= r ∈ R, there exists a unit u ∈ R and unique integers fi , 0 ≤ fi ≤ ei , such that r = uβ1f1 · · · βsfs . Proof. Suppose Φ(r) = (r1 , · · · , rs ). Now let ri 6= 0 for all i, then ri = ui γifi with 0 ≤ fi ≤ ei − 1 by Lemma 4.2, and this gives that Φ(r) = = = =

(r1 , · · · , rs ) = (u1 γ1f1 , · · · , us γsfs ) = (u1 , · · · , us )(γ1f1 , · · · , γsfs ) (u1 , · · · , us )(γ1f1 , · · · , 1) · · · (1, · · · , γsfs ) (u1 , · · · , us )(γ1 , · · · , 1)f1 · · · (1, · · · , γs )fs Φ(u)Φ(β1f1 ) · · · Φ(βsfs ) = Φ(uβ1f1 · · · βsfs ),

where u is the inverse image of (u1 , · · · , us ). Since Φ is an isomorphism, we get that r = uβ1f1 · · · βsfs . If ri = 0 for some i, then r ∈ Ker(Φi ), this implies that r = γβiei . This can be done where γ is not a multiple of βi giving the uniqueness in this case. Let A = {i | Φi (r) = 0} then Q r = β i∈A βiei with Φj (β) 6= 0 for all j. Then proceed with β as before, i.e., β = uβ1f1 · · · βsfs . Q Q Because βiei +1 = βiei , we have that r = u i6∈A βifi i∈A βiei . Theorem 4.4. Let R be a finite principal ideal ring. If C is a code of length n over R then C∼ = R/hd1 i ⊕ R/hd2 i ⊕ · · · ⊕ R/hdr i,

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where di is not a unit for all i, and hd1 i ⊇ hd2 i ⊇ · · · ⊇ hdr i (r ≤ n) are uniquely determined principal ideals in R. Proof. The proof of Theorem 4.4 can be obtained from results in [4]. The result follows from Proposition 25.4.6B and Theorem 25.3.3 in [4]. Notice that if R = F is a field, then C ∼ = F r , where r is the dimension as we would like. 13

Now suppose φ : C → R/hd1 i ⊕ R/hd2 i ⊕ · · · ⊕ R/hdr i is the isomorphism given by Theorem 4.4. t Let hdj i be an ideal in R. By Theorem 4.4 and Lemma 4.3 there exists βi j ∈ R, tj ≤ ej , t such that βi j dj . Then this gives a surjective homomorphism: t

t

R/hdj i → Ri /hγi j i, a + hdj i 7→ Φi (a) + hγi j i.

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To verify that this homomorphism is well-defined, suppose a + hdj i = a0 + hdj i, then a − a0 ∈ t hdj i, and this implies that a − a0 = aj for some aj ∈ hdj i. Then aj = rdj = rr0 βi j for some r, r0 ∈ R. Hence t

t

t

Φi (a) − Φi (a0 ) = Φi (a − a0 ) = Φi (aj ) = Φi (rr0 βi j ) = Φi (rr0 )γi j ∈ hγi j i, t

t

and this gives Φi (a) + hγi j i = Φi (a0 ) + hγi j i. Thus the correspondence in (17) is well-defined. Notice that Φi is a surjective homomorphism, and this gives directly that this function is also a surjective homomorphism. For simplicity, we denote the homomorphism in (17) by Ψi . It can be easily extended to Ψi : R/hd1 i ⊕ R/hd2 i ⊕ · · · ⊕ R/hdr i → Ri /hγit1 i ⊕ Ri /hγit2 i ⊕ · · · ⊕ Ri /hγitr i, by (a1 + hd1 i, a2 + hd2 i, · · · , ar + hdr i) 7→ (Φi (a1 ) + hγit1 i, Φi (a2 ) + hγit2 i, · · · , Φi (ar ) + hγitr i). Therefore, we have Ψi (R/hd1 i ⊕ R/hd2 i ⊕ · · · ⊕ R/hdr i) = Ri /hγit1 i ⊕ Ri /hγit2 i ⊕ · · · ⊕ Ri /hγitr i. Suppose that Φi (u) = Φi (v) for u, v ∈ C, then u − v ∈ Ker(Φi ). By Lemma 4.1, this implies that u = v + ββiei w for some β ∈ R and w ∈ Rn . So φ(u) = φ(v) + ββiei φ(w). This gives that Ψi (φ(u)) = Ψi (φ(v)) + Ψi (ββiei φ(w)) = Ψi (φ(v)). So we have a well-defined map from Ci to Ri /hγit1 i ⊕ Ri /hγit2 i ⊕ · · · ⊕ Ri /hγitr i, denoted by φi , φi : Ci → Ri /hγit1 i ⊕ Ri /hγit2 i ⊕ · · · ⊕ Ri /hγitr i, Φi (u) 7→ Ψi (φ(u)).

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φi (Φi (u)) = Ψi (φ(u)).

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That is I.e., the following diagram commutes: φ

C −−−→   Φi y

R/hd1 i ⊕ R/hd2 i ⊕ · · · ⊕ R/hdr i  Ψ y i

φi

Ci −−−→ Ri /hγit1 i ⊕ Ri /hγit2 i ⊕ · · · ⊕ Ri /hγitr i. Following the discussion above, we have the following theorem. 14

Theorem 4.5. Assume the notations given above. The map in (18) is a surjective homomorphism. Proof. The fact that Ψi and φ are both surjective homomorphisms gives that φi is a surjective homomorphism. Theorem 4.6. Suppose φ : C → R/hd1 i ⊕ R/hd2 i ⊕ · · · ⊕ R/hdr i as before. Let vj be the codeword in C corresponding to (0, · · · , 1, · · · , 0) in the direct product, where 1 ∈ R/hdj i is in the j-th coordinate position. Then v1 , v2 , · · · , vr form a basis for C. Proof. It is clear that v1 , v2 , · · · , vr generate C, since the set {(1, · · · , 0, · · · , 0), . . . , (0, · · · , 1, · · · , 0), . . . , (0, · · · , 0, · · · , 1)} generates R/hd1 i⊕R/hd2 i⊕· · ·⊕R/hdr i, and φ is an isomorphism. In order to prove they are P P independent, suppose aj vj = 0. Then φ( aj vj ) = (a1 , a2 , · · · , ar ) = 0, which implies that aj ∈ hdj i for all j. Since the image of vj under the isomorphism φ is (0, · · · , 1, · · · , 0), we have that φ(aj vj ) = aj φ(vj ) = aj (0, · · · , 1, · · · , 0) = (0, · · · , aj , · · · , 0) = (0, · · · , 0, · · · , 0) = 0. This means that aj vj = 0, since φ is an isomorphism. Since d1 is not a unit, there exists βi for some i such that βi d1 , where βi is the inverse image of γi . We prove that Φi (v1 ), Φi (v2 ), · · · , Φi (vr ) are modular independent over Ri . P Suppose al Φi (vl ) = 0. Then there exist bl ∈ R such that Φi (bl ) = al , since Φi is a surjective homomorphism from R to Ri . Then X X X al Φi (vl ) = Φi (bl )Φi (vl ) = Φi ( bl vl ) = 0. This implies that X

bl vl = ββiei w, for some w ∈ Rn and β ∈ R.

Hence (b1 , b2 , · · · , br ) =

X

bl (0, · · · , 1, · · · , 0) =

X

X bl φ(vl ) = φ( bl vl ) = ββiei φ(w).

So Ψi (b1 , b2 , · · · , br ) = Ψi (ββiei φ(w)) = 0. This implies that (b1 , b2 , · · · , br ) ∈ Ker(Ψi ). This result gives that the identity above implies that al = Φi (bl ) ∈ hγitl i ⊆ hγi i for all l, since tl ≥ 1. This implies that Φi (v1 ), Φi (v2 ), · · · , Φi (vr ) 15

are modular independent. Recall that the rank of a code is the minimum number of generators of the code. Now we can prove the following result. Theorem 4.7. Let R be a principal ideal ring and let C be an arbitrary code over R, then any basis for C contains exactly r codewords, where r is the rank of C. Proof. Theorem 4.6 gives that a basis exists for C, let w1 , w2 , · · · , wr be that basis. There exists an index i such that Φi (C) has rank r, since it is well known (see [1]) that the rank of C is the maximum of the ranks of Cj = Φj (C). If v1 , v2 , · · · , vs is also a basis, then Φi (v1 ), Φi (v2 ), · · · , Φi (vs ) generate Ci over Ri , so s ≥ r. Since basis vectors are modular independent, there exists j such that Φj (v1 ), Φj (v2 ), · · · , Φj (vs ) are modular independent. This gives that s ≤ rank(Cj ) by using Lemma 3.1 and the form of the generator matrices for chain rings given in [11]. We know rank(Cj ) ≤ r which gives s ≤ r. This gives s = r and hence any basis for C contains exactly r codewords. Acknowledgment: We are grateful to Sergio L´opez-Permouth for helpful discussions on rings and for pointing out reference [4].

References [1] Dougherty, S. T., Kim, J-L., Kulosman, H. , MDS codes over finite principal ideal rings, to appear in Designs, Codes and Cryptography. [2] Dougherty, S. T., Park, Y. H., On modular cyclic codes, Finite Fields and their Applications, Vol. 13, No. 1, 31-57, 2007. [3] Dougherty, S. T., Gulliver, T. A., Park, Y. H., Wong, J. N. C., Optimal linear codes over Zm , Journal of the Korean Mathematical Society, Vol. 44, 1139-1162, 2007. [4] Faith, C., Algebra II: Ring Theory, Grundlehren Math. Wiss, Bd.191, Springer, Berlin,1976. [5] Hammons, A. R., Jr., Kumar, P. V., Calderbank, A. R. , Sloane, N. J. A. , Sol´e, P., The Z4 linearity of Kerdock, Preparata, Goethals and related codes, IEEE-IT, Vol. 40, 301-319, 1994. [6] Horimoto, H., Shiromoto, K., A Singleton bound for linear codes over quasi-Frobenius rings, Proceedings of the 13th International Symposium on Applied Algebra, Algebraic Algorithms, and Error-Correcting Codes, Hawaii (USA), 51-52, 1999. 16

[7] Jacobson, N., Basic Algebra I, W H Freeman & Co., 1985. [8] MacWilliams, F. J., Sloane, N. J. A., The Theory of Error-Correcting Codes, NorthHolland, Amsterdam, 1977. [9] Matsumura, H., Commutative Ring Theory, Cambridge University Press, Cambridge, 1989. [10] McDonald, D., Finite rings with identity, Marcel Dekker, New York, 1974. [11] Norton, G. H., Sˇalˇagean, A., On the structure of linear and cyclic codes over a finite chain ring, Applicable Algebra in Engineering, Communication and Computing, Vol. 10, 489-506, 2000. [12] Park, Y. H., Modular independence and generator matrices for codes over Zm , to appear in Designs, Codes and Cryptography. [13] Wood, J., Duality for modules over finite rings and applications to coding theory. American Journal of Mathematics, Vol. 121, 555-575, 1999.

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