Cyclic codes over Rk Steven T. Dougherty Department of Mathematics University of Scranton Scranton, PA 18510 USA e-mail: [email protected] Suat Karadeniz Bahattin Yildiz Department of Mathematics Fatih University 34500 Istanbul, Turkey e-mail: [email protected] e-mail: [email protected] June 22, 2011 Abstract Cyclic codes over an infinite family of rings are defined. We prove that the binary images of cyclic codes over these rings under the natural Gray map are binary quasicyclic codes of degree 2k . Further, we construct several optimal or near optimal binary codes via this map.

1

Introduction

Cyclic codes are an important class of linear codes and have generated great interest in coding theory. In general, they have a natural encoding and decoding algorithm. Moreover, since they can be described as as ideals in some polynomial rings, they have a rich algebraic structure. There is a vast literature on cyclic codes over fields and more recently over rings. For some of these we can refer to [2], [4], [5], [6], [8], [9], [15], and [23]. In particular, cyclic codes 1

over the rings of order 4 are of interest since they all have a Gray map to binary codes. Cyclic codes over Z4 were studied in a series of papers, [1],[3], [8], [14], [16], [17], and [18]. Cyclic codes over the ring F2 + uF2 were studied in [2] and [4]. Cyclic codes over a generalization of this ring, namely R2 = F2 + uF2 + vF2 + uvF2 were then introduced by Yildiz and Karadeniz in [23], in which the authors gave a partial characterization of cyclic codes over the ring R2 by using a homomorphism from R2 to R1 = F2 + uF2 inspired from [2]. In the present work we shall study cyclic codes over an infinite family of rings that generalize these rings. This infinite family of rings, Rk , together with the properties of linear codes over these rings, were introduced in [10]. The paper is organized as follows. In Section 2, we will give the necessary background on the rings Rk and on linear and cyclic codes over Rk , referring to many of the results in [10]. In Section 3, we will describe cyclic codes over Rk using the decompositions of the polynomial xn − 1 over the ring Rk . The ideals of the ring Rk [x]/(xn − 1) will be described and a formula for the number of cyclic codes over Rk of certain lengths will be given. Also, necessary and sufficient conditions on n will be given for Rk [x]/(xn − 1) to be a local ring. In Section 4, we will classify one-generator cyclic codes over the ring Rk . In particular, we will obtain necessary and sufficient conditions on the coefficients of a polynomial in Rk [x]/(xn −1) to generate a non-trivial cyclic code. Later we will use these ideas to construct some specific one-generator cyclic codes over Rk whose binary images are optimal or nearoptimal. In Section 5, we will describe the images of cyclic codes over Rk . In the first part, we will consider the Rk−1 images of odd length cyclic codes over Rk , and we will prove that they are equivalent under the Nechaev permutation to cyclic codes. In the second part, we will prove that the binary images of cyclic codes over Rk under the Gray map ψk are actually binary quasi-cyclic codes of degree 2k . We will conclude with some remarks and directions for possible future work.

2

Basics

Define the following ring for k ≥ 1. Let Rk = F2 [u1 , u2 , . . . , uk ]/hu2i = 0, ui uj = uj ui i. For all k, the ring Rk is a finite commutative ring. For any subset A ⊆ {1, 2, . . . , k} let Y uA := ui (1) i∈A

with the convention that u∅ = 1. This gives that any element of Rk can be represented as X cA uA , c A ∈ F2 . (2) A⊆{1,...,k}

2

We recall that a local ring is a ring with a unique maximal ideal. It follows immediately k that the ring Rk is a local ring with maximal ideal hu1 , u2 , . . . , uk i and |Rk | = 2(2 ) . This ring is neither a principal ideal ring nor a chain ring when k ≥ 2. The ring is, however, a Frobenius ring. In [21], it is shown that codes over Frobenius rings satisfy both MacWilliams theorems. See [21] for foundational results on codes over Frobenius rings. It is known that an element of Rk is a unit if and only if the coefficient of u∅ is 1 and that each unit is also its own inverse. See [10] for proofs of these and other results. We say that a linear code of length n over Rk is an Rk -submodule of Rkn . We define the Gray map inductively, extending it naturally from the Gray map on R1 from [7] as follows. n , then we can define For c ∈ Rkn , we can write c = c1 + uk c2 with c1 , c2 ∈ Rk−1 φk (c) = (φk−1 (c2 ), φk−1 (c1 ) + φk−1 (c2 )) , with φ0 being the identity map on F2 . The Lee weight of a codeword is the Hamming weight of the image of the codeword k under φk . Then the Gray map is a linear weight preserving map from Rkn to F22 n . It is immediate that φk is one-to-one and that wL (uA ) = 2|A| for each A ⊆ {1, 2, . . . , k}, see [10] for details. The Hamming weight of a vector c is denoted wt(c) and is the number of non-zero coordinates of the element. The minimum weight is the minimum of all non-zero weights in the code. We denote the minimum Hamming distance by dH (C) and the minimum Lee distance by dL (C). For a code C of length n over Rk , we can define the torsion and the residue codes in a similar way that was done in [7]:  n T or(C) = a ∈ Rk−1 |uk a ∈ C , and n Res(C) = {a ∈ Rk−1 | ∃b : a + uk b ∈ C}.

Note that T or(C) and Res(C) are both linear codes over Rk−1 if C is linear over Rk , of length n. Now, if we let ϕ : C → Res(C) be the map ϕ(a + uk b) = a, we clearly see that ϕ is a well defined map that is onto with ker(ϕ) = T or(C). This means |C| = |T or(C)| · |Res(C)|.

(3)

A cyclic shift on Rkn is a permutation τ such that τ (c0 , c1 , · · · , cn−1 ) = (cn−1 , c0 , c1 , · · · , cn−2 ).

(4)

A linear code C over Rk of length n is said to be a cyclic code if it is invariant under the cyclic shift, i.e., τ (C) = C. Note that by the above definition, it is easy to see that if C is cyclic, then both T or(C) and Res(C) are cyclic codes. 3

Using the natural polynomial representation of codewords in Rkn in Rk [x], that is the P vector (a0 , a1 , . . . , an−1 ) corresponds to the polynomial ai xi , we see that for a codeword c ∈ Rkn , τ (c) corresponds to xc(x) in Rk [x]/(xn − 1). The following is an easy extension of the characterization of cyclic codes over R1 . Proposition 2.1. A subset C of Rkn is a linear cyclic code of length n over Rk if and only if its polynomial representation is an ideal of the ring Rk,n := Rk [x]/(xn − 1).

3

Cyclic Codes

We first note that, unlike Z2k , the ring F2 is a subring of Rk for all k. This makes the description of generator polynomials much simpler. Define the map Πt : Rt → Rt−1 where α ∈ Rt is mapped to α (mod ut ). Let µk = Π1 ◦ P Π2 ◦ · · · ◦ Πk . Note that in the notation from Section 2, µk takes an element A⊆{1,...,k} cA uA to c∅ . Apply the map to polynomials as follows X X µk (p(x)) = µk ( α i xi ) = µk (αi )xi . A monic polynomial in Rk [x] is said to be a basic irreducible polynomial if its projection under µk is an irreducible polynomial in F2 [x]. Denote by GR(Rk , s) = Rk [x]/hp(x)i where p(x) is a basic monic irreducible polynomial of degree s. If p(x) ∈ F2 [x] then p(x) can be viewed as an element of Rk [x] as well. Hence, if p(x) is irreducible in F2 [x] then it is irreducible in Rk [x]. This is only possible since F2 is a subring of Rk . Let p(x) be an irreducible polynomial over F2 of degree s. Then F2 [x]/hp(x)i is a field of order 2s . Let A0 = F2 [x]/hp(x)i and define At = At−1 [ut ]/hu2t = 0, ut ui = ui ut i.

(5)

Any element in At is of the form α + βut where α and β are in At−1 . Lemma 3.1. The element α + βut in At is a unit if and only if α is a unit in At−1 . Proof. Let α + βut , γ + δut ∈ At , then (α + βut )(γ + δut ) = αγ + (βγ + αδ)ut . If α is a unit then let γ = α−1 and δ = α−1 βγ. Then the product is 1. Therefore, if α is a unit then α + βut is a unit for any β. If α is not a unit then it is a zero divisor since the ring is finite. Let  be such that α = 0. Then ut (α + βut ) = ut α + βtu2t = 0. Hence α + βut is a zero divisor for all β.

4

Let U(At ) be the group of units of the ring At . By the previous lemma we have that |U(At )| = |U(At−1 )||At−1 |.

(6)

We know that |A0 | = 2s and |U(A0 )| = 2s − 1. This gives that |U(A1 )| = 2s (2s − 1). This result for k = 1 appears in [4]. t Notice also that |At | = (2s )2 since there are 2t subsets of {u1 , . . . , ut } and 2s choices for each coefficient. Then by induction we get the following. Theorem 3.2. The group of units U(At ) is the direct product of a cyclic group G of order t t t 2s − 1 and an abelian group H of order (2s )2 −1 , with |U(At )| = (2s )2 −1 (2s − 1) = (22 )s − t (22 −1 )s . It follows from this theorem that elements of the group H are of the form 1 + αA uA where A ⊆ {1, 2, . . . , k} and αA ∈ G. It follows that the zero divisors in Rk [x]/hp(x)i are of P the form αA uA , αA ∈ A0 , α∅ = 0. Notice that |A0 | = 2s and there are 2t − 1 subsets of {1, 2 . . . , t} other than the emptyset. t t So there are (2s )2 −1 = (22 −1 )s non-units in At . We collect these results in the following theorem. t

Theorem 3.3. The ring At has cardinality |At | = (22 )s . The units correspond to elements P t t of the form αA uA , αA ∈ A0 , α∅ = 1 and there are (22 )s − (22 −1 )s of them. The non-units P t are of the form αA uA , αA ∈ A0 , α∅ = 0 and there are (22 −1 )s of them. This gives the following important corollary. Corollary 3.4. The ideals of At bijectively correspond to the ideals of Rt . That is, any ideal in At is of the form hβ1 , β2 , . . . , β` i, where βi ∈ Rt . The following generalizes a result in [4]. Q Lemma 3.5. If xn − 1 = pi , where pi ∈ Rk [x] are basic irreducible and pairwise coprime and n is odd, then the factorization is unique. Proof. The field F2 is a subring of Rk and xn − 1 factors uniquely as a product of pairwise coprime irreducible polynomials in F2 [x]. Thus it factors over Rk since F2 is a subring of Rk . Recall that Rk is a local ring with unique maximal ideal hu1 , u2 , . . . , uk i (see [10] for details). Thus Hensel’s Lemma gives that regular polynomials (polynomials that are not zerodivisors) over Rk have a unique factorization. Theorem 3.6. Let n be odd and let xn − 1 = p1 p2 . . . pr . The ideals in Rk [x]/hxn − 1i are of the form I ∼ = I1 ⊕ I2 ⊕ · · · ⊕ Ir where Ij is an ideal of Rk [x]/hpj i. These ideals correspond to the ideals of Rk as in Corollary 3.4. 5

Corollary 3.7. Let Ik be the number of ideals in Rk and let t be the number of basic irreducible polynomial factors in xn − 1. Then the number of cyclic codes is (Ik )t . Over Rk , k ≥ 2 there are many more cyclic codes to consider than over R1 since R1 has only three ideals, namely h0i, h1i and hui whereas Rk has numerous ideals. Hence the lifting of the factorization of F2 gives a substantial number of cyclic codes. Note that it is quite a challenging question to determine the number of ideals in Rk . For example, for k = 2, we know that the answer is 7 from [22]. But in general we do not even know the answer for the case k = 3. This question which is related to cyclic codes can be considered for future works in this area. We finish this section by considering some further properties of the ring Rk,n = Rk [x]/(xn − 1). We know that when k = 1, the ring R1 is a chain ring and hence when n is odd R1,n is also a chain ring([6]). However when k ≥ 2, Rk,n is not a chain ring as for example the ideals < u1 >, < u2 > in Rk,n are not related via any inclusion. Similarly one can easily show that Rk,n is not a principal ideal ring. However we can prove the following important property of Rk,n when n = 2m is a power of 2. Theorem 3.8. Rk,n is a local ring when n = 2m for some m ∈ Z+ . Proof. It is enough to show that the set of all non-units in Rk,n forms an ideal. Since a non-unit multiplied by any element is again a non-unit, to complete the proof we need to show that the set of non-units in Rk,n is an additive subgroup of Rk,n . Now let α = a0 + a1 x + · · · + a2m −1 x2

m −1

(7)

m

be an element of Rk,n . Since char(Rk ) = 2 and x2 = xn = 1 in Rk,n , we see that m

m

m

m

α2 = a20 + a21 + · · · + a22m −1 .

(8)

But, note that in Rk , a2 = 0 or 1 depending on whether a is non-unit or unit in Rk . Now, m m if α2 = 0, then α is a non-unit while if α2 = 1 then α is a unit. Thus, we get a nice m characterization of units in Rk,2m , namely, α = a0 + a1 x + · · · + a2m −1 x2 −1 ∈ Rk,2m is a non-unit if and only if an even number of ai ’s are units. But in Rk , the sum of two non-units is a non-unit, the sum of a non-unit and a unit is a unit and the sum of two units is a m non-unit. This means that the elements α = a0 + a1 x + · · · + a2m −1 x2 −1 ∈ Rk,2m with an even number of units form an additive subgroup of Rk,2m , which completes the proof. To prove that the result in the above theorem is exclusive, we introduce γ : Rk,n → Rk by γ(a0 + a1 x + · · · an−1 xn−1 ) = a0 + a1 + · · · + an−1 . Note that one can easily show that γ is a surjective ring homomorphism (in fact as we’ll see below, with the connection of cyclic codes and group rings, γ corresponds to the well-known augmentation homomorphism). Now we are ready to prove the following result: 6

Theorem 3.9. The ring Rk,n is not a local ring when n = 2m · s with s > 1 an odd number. Proof. Again it is enough to show that the set of non-units in Rk,n does not form an ideal when n = 2m · s with s > 1. First, observe that m ·s

0 = x2

m

+ 1 = (x2 + 1)(x2

m (s−1)

m (s−2)

+ x2

m

+ · · · + x2 + 1)

in Rk,2m ·s . So, f1 (x) = x2

m (s−1)

m (s−2)

m

+ x2

+ · · · + x2 + 1

(9)

is a zero-divisor and hence a non-unit in Rk,2m ·s . On the other hand when s > 1 is an odd number we have m (s−2)

γ(x2

m

+ · · · + x2 + 1) = 1 + 1 + · · · + 1 = s − 1 = 0

in Rk . Since γ is a surjective ring homomorphism, we see that m (s−2)

f2 (x) = x2

m

+ · · · + x2 + 1

(10)

is also a non-unit in Rk,2m ·s . But then f1 (x) + f2 (x) = x2

m (s−1)

,

which is a unit in Rk,2m ·s because m

m (s−1)

x 2 · x2

m ·s

= x2

=1

in Rk,2m ·s . This completes the proof that Rk,2m ·s cannot be a local ring when s > 1 is odd.

4 4.1

One generator cyclic codes over Rk Structure of one-generator cyclic codes

We give a full characterization of one generator cyclic codes over Rk . We let p(x) = a0 + a1 x + · · · + an−1 xn−1 be a polynomial in Rk [x] and C be the cyclic code of length n obtained as the ideal generated by p(x) in Rk,n . Of course, if p(x) is a unit in Rk,n , then we can easily see that the cyclic code C is the trivial code Rkn . Therefore, in order to get interesting cyclic codes over Rk , we have to choose p(x) as a non-unit in Rk,n . This is actually an interesting problem to consider. The case when k = 2 was done in [23], but the conditions obtained were not in a simple form. Our aim here is to give best possible characterization for the coefficients ai that would make p(x) a non-unit in Rk,n as well as any properties of such codes. 7

We know that Rk,n ' Rk G where G =< g : g n = 1 > is the cyclic group of order n and Rk G denotes the group ring. The isomorphism is quite obvious in that we map a0 +a1 x+· · ·+an−1 xn−1 to a0 +a1 g+. . . an−1 g n−1 . Now, for Rk G we have a nice representation by matrices. For the results in this section we refer to [13]. To every element in Rk G, and hence in Rk,n , corresponds a circulant matrix in the form:   a0 a1 a2 . . . an−1    an−1 a0 a1 . . . an−2     . . . . . . .  n−1   σ(a0 + a1 x + . . . an−1 x ) =  (11)  . . . . . . .     . . . . . .   . a1 a2 a3 . . . a0 An immediate consequence of this description is the characterization of the units and zerodivisors in Rk,n . Note that the determinant function det is a multiplicative map from matrices over a commutative ring R to the ring R. The results in [13] for general group rings imply that α ∈ Rk,n is a unit if and only if det(σ(α)) is a unit in Rk . This helps us obtain the following corollary: Corollary 4.1. An element α = a0 + a1 x + . . . an−1 xn−1 is a unit in Rk,n if and only if det(σ(α)) is a unit in Rk . Equivalently α is a non-unit in Rk,n if and only if det(σ(α)) ∈ Iu1 ,u2 ,··· ,uk . This looks like a good characterization, however in general, we can improve on this characterization. Recall that the map µk : Rk → F2 was defined as follows in section 3:   X µk  cA uA  = c∅ . (12) A⊆{1,...,k}

Essentially µk reduces the elements of Rk modulo u1 , u2 , · · · , uk . Evidently, µk maps every unit in Rk to 1 and every non-unit to 0. Note that because of the properties of units and non-units in Rk , we see that µk must be a ring epimorphism. Recall also that µk acts on polynomials as well by just acting on the coefficients. And since it is a ring homomorphism, we easily observe that det(σ(µk (p(x)))) = µk (det(σ(p(x)))). (13) This means that p(x) = a0 + a1 x + . . . an−1 xn−1 is a non-unit in Rk,n if and only if the {0, 1}-matrix σ(µk (p(x))) obtained by taking each ai to µk (ai ) is a singular matrix in Fn×n . 2 Note that the matrix σ(p(x)) is uniquely determined by p(x).

8

Let’s give an example in R3 . Let n = 4 and p(x) = 1+(u1 +u2 )x+(1+u1 u3 +u1 u2 u3 )x2 + u2 x3 . Then µk (p(x)) = 1 + x2 and so we will have   1 0 1 0  0 1 0 1    σ(µk (p(x))) =    1 0 1 0  0 1 0 1 which is singular and hence we see that p(x) is non-unit in R3,4 . On the other hand it is easy to see that p(x) · p(x) = 0 in R3,4 , so that it is indeed a non-unit. Another characterization of non-trivial one-generator cyclic codes can be given by looking at µk (p(x)). Note that, by the above observation we know that p(x) generates a non-trivial cyclic code C over Rk if and only if σ(µk (p(x))) is singular. But this means that the binary cyclic code µk (C) generated by µk (p(x)) is non-trivial. This is true if and only if µk (p(x)) is a non-unit in F2 [x]/(xn − 1). But we know that the polynomial µk (p(x)) ∈ F2 [x] is a unit in F2 [x]/(xn − 1) if and only if GCD(µk (p(x)), xn − 1) = 1. Thus we have proven: Theorem 4.2. Let p(x) = a0 + a1 x + · · · + an−1 xn−1 ∈ Rk [x]. Then C =< p(x) > is a non-trivial cyclic code if and only if GCD(µk (p(x)), xn − 1) 6= 1. We give some examples of constructions of one-generated cyclic codes below, where we take Theorem 4.2 into account. Before we proceed with the examples, we can make the following bound for the minimum distance of one-generator cyclic codes: Theorem 4.3. Let C =< p(x) > be a one-generator cyclic code over Rk of length n. By wL (p(x)) we mean the Lee weight of the coefficient vector of p(x) and we similarly define wH (µk (p(x))) as the Hamming weight of the binary coefficient vector of µk (p(x)). Then, with dL (C) denoting the minimum Lee distance of C, we have dL (C) ≤ min{wL (p(x)), 2k wH (µk (p(x)))}. Proof. Since p(x) ∈ C, we have that dL (C) ≤ wL (p(x)). On the other hand, we know, from the properties of the ring Rk that (u1 u2 · · · uk ) · p(x) = (u1 u2 · · · uk ) · µk (p(x)). Then since u1 u2 · · · uk p(x) ∈ C, we have that u1 u2 · · · uk µk (p(x)) ∈ C, so that we have dL (C) ≤ wL (u1 u2 · · · uk µk (p(x))) = 2k wH (µk (p(x))).

These inequalities are easily observed in the examples below. 9

4.2

Examples of good binary codes obtained from one-generator cyclic codes over Rk

Below we give some results concerning optimal or near optimal binary codes obtained as the Gray images of cyclic codes over R2 and R3 . In the tables below, [.]∗ will denote that the binary code is an optimal one, [.]b will denote that the binary code has the best known minimum distance for those parameters, while [.]∗−t and [.]b−t will denote that the binary code has minimum distance that is t less than the optimal and best known code, respectively. Finally [.]sd will denote that the cyclic code is at the same time self-dual and is extremal. We first start with R2 : Table 1: One-generator cyclic codes of length 4 over R2 and binary images f φ2 (< f >) 2 3 (u1 u2 + u1 + u2 ) + (u1 u2 + u1 + u2 )x + (u1 + u2 )x + (u1 + u2 )x [16, 3, 8]∗ u1 u2 + (u1 + u1 u2 )x + u1 x3 [16, 4, 8]∗ (u1 u2 + u1 + 1) + (u2 + 1)x + (u1 + 1)x2 + (u1 u2 + u2 + 1)x3 [16, 5, 8]∗ (u1 + u2 + 1) + (u1 u2 + u2 + 1)x + (u2 + 1)x2 + (u1 + u2 + 1)x3 [16, 6, 6]∗ (u1 u2 + 1) + (u1 + u2 + 1)x + (u1 u2 + 1)x2 + (u1 + 1)x3 [16, 8, 4]sd

Table 2: One-generator cyclic codes of length 5 over R2 and binary images f (u1 u2 + u2 ) + (u1 u2 + u1 )x + u2 x2 + (u1 + u2 )x3 + (u1 u2 + u1 + u2 )x4 (u1 u2 + u1 + u2 + 1) + (u1 u2 + 1)x + (u2 + 1)x2 +( u1 u2 + 1)x3 + (u1 + u2 + 1)x4 (u1 + u2 ) + (u1 u2 + u1 + u2 )x + u1 x2 + (u1 u2 + u1 )x4

φ2 (< f >) [20, 10, 4]sd [20, 12, 4]∗ [20, 8, 8]∗

Table 3: One-generator cyclic codes of length 6 over R2 and binary images f φ2 (< f >) 2 3 5 u1 u2 (1 + x + x + x ) [24, 2, 16]∗ u1 + u2 + u2 x + u1 x2 + (u1 u2 + u1 + u2 )x3 + u2 x4 + (u1 u2 + u1 )x5 [24, 4, 12]∗ u1 x + u1 u2 x2 + (u1 u2 + u1 + u2 )x3 + u2 x4 + (u1 u2 + u1 + u2 )x5 [24, 8, 8]∗ u1 + (u1 u2 + u1 )x + (u1 u2 + u1 + u2 )x2 + u1 u2 x3 + u2 x4 + (u1 u2 + u1 )x5 [24, 9, 8]∗ (u1 u2 + u1 ) + (u1 u2 + u2 )x2 + (u1 + u2 )x3 + (u1 u2 + u1 )x4 + u1 x5 [24, 10, 8]∗ 1 + (u1 u2 + u1 + 1)(x + x5 ) + (u1 u2 + u1 + u2 + 1)x2 + x3 + (u2 + 1)x4 [24, 12, 6]∗−2 (u1 u2 + u2 + 1) + (u1 + 1)x + x2 + (u1 + u2 )(x2 + x4 + x5 ) + u2 x3 + u1 u2 x4 [24, 16, 3]∗−1

10

Table 4: One-generator cyclic codes of length 7 over R2 and binary images f φ2 (< f >) 2 4 u1 u2 (1 + x + x + x ) [28, 3, 16]∗ u1 + u2 x + u2 x2 + (u1 + u2 )x3 + u2 x5 + (u1 + u2 )x6 [28, 6, 12]∗ u1 + u2 x + u1 u2 (x2 + x4 + x5 ) + (u1 + u2 )(x3 + x5 ) + u2 x2 + u1 x4 [28, 7, 12]∗ u1 u2 (1 + x + x4 + x6 ) + u2 (1 + x + x3 ) + u1 (x + x2 + x3 + x6 ) [28, 8, 10]∗−1 (u1 u2 + u1 )(1 + x3 + x6 ) + u1 u2 x + (u1 + u2 )x4 + u2 x6 [28, 12, 8]∗

Table 5: One-generator cyclic codes of length 8 over R2 and binary images f φ2 (< f >) 4 5 6 7 3 6 7 2 3 4 2 u1 u2 (1 + x + x + x + x + x ) + u2 (1 + x + x + x ) + u1 (x + x + x ) + x [32, 16, 8]sd u1 + u1 u2 (x + x3 ) + (u1 u2 + u1 )(x2 + x4 ) + u1 x6 [32, 5, 16]∗ (u1 + u2 + u1 u2 )(1 + x4 + x5 ) + (u1 u2 + u1 )(x2 + x6 + x7 ) + (u1 + u2 )x + u1 x3 [32, 6, 16]∗ u1 x + (u1 u2 + u2 + 1)(x + x2 + x7 ) + (u2 + 1)(1 + x3 + x4 ) + x5 + (u1 + 1)x6 [32, 14, 8]b If we take n = 10 and f (x) = u1 u2 + u1 x + u1 x2 + u1 x3 + u1 x4 + u1 u2 x5 + (u1 u2 + u1 )x6 + u1 x7 + u1 x8 + (u1 u2 + u1 )x9 we get φ2 (< f >) as a binary linear code of parameters [40, 8, 16], which is optimal. If we take n = 10 again and this time f (x) = u2 (1 + x2 + x4 + x6 + x8 ) + u1 (x + x3 + x5 + x7 + x9 ), we get φ2 (< f >) as a binary linear code of parameters [40, 4, 20], which is optimal. Taking n = 14 and f (x) = u1 u2 (1 + x + x3 + x6 + x7 + x8 + x10 + x13 ), we get φ2 (< f >) to be a binary linear code with parameters [56, 3, 32] which is also optimal. Finally, for n = 14 again, if we take f (x) = u1 +u2 +(u1 u2 +u1 +u2 )x+u1 x2 +(u1 u2 +u1 )x3 +(u1 u2 +u1 )x4 +u2 x5 +(u1 +u2 )x7 + (u1 u2 + u2 )x8 + (u1 + u2 )x9 + u1 u2 x10 + u2 x11 + (u1 u2 + u1 )x12 + u2 x13 , we get φ2 (< f >) to be a binary linear code with parameters [56, 18, 16] which is the best known code of that size. Now, let’s look at some examples over R3 :

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Table 6: One-generator cyclic codes of length 2 over R3 and binary images f φ3 (< f >) u2 + u2 u3 + (u1 u2 + u2 + u3 )x [16, 8, 4]∗ u1 + u3 + u1 u3 + u1 u2 u3 + (u1 + u3 )x [16, 4, 8]∗ u1 u3 + u2 u3 + u3 + (u1 u2 u3 + u1 u2 + u1 u3 + u3 )x [16, 6, 6]∗

Table 7: One-generator cyclic codes of length 4 over R3 and binary images f φ3 (< f >) (u1 u2 u3 + u1 u2 + u1 u3 + u1 + u2 u3 + u2 ) + (u1 u2 + u1 + u2 u3 + u3 )x + (u1 u2 + u1 u3 + u1 + u2 u3 + u3 )x2 + (u1 u2 u3 + u1 u2 + u1 u3 + u3 )x3 [32, 16, 8]∗ (u1 u2 u3 + u1 u2 + u2 u3 + u3 ) + (u1 u2 u3 + u1 u2 + u1 u3 + u1 + u2 u3 + u2 + u3 )x + (u1 u2 + u1 u3 + u1 + u2 u3 )x2 + (u1 u2 + u1 u3 + u2 )x3 [32, 14, 8]b (u1 u3 + u2 u3 + u2 ) + (u1 u2 u3 + u1 u3 + u2 u3 + u2 )x + (u1 u2 u3 + u2 )x2 + (u1 u2 u3 + u1 u2 + u2 )x3 [32, 8, 12]∗−1

Table 8: One-generator cyclic codes of length 5 over R3 and binary images f φL (< f >) (u1 u2 + u1 + u2 u3 + u2 + u3 ) + (u1 u2 u3 + u1 u2 + u1 u3 + u2 u3 + u3 )x+ (u1 u2 u3 + u1 u2 + u1 u3 + u2 )x2 + (u1 u2 u3 + u1 u3 + u1 + u2 )x3 + (u1 u2 + u1 u3 + u2 u3 + u2 )x4 [40, 18, 8]b−2 (u1 u2 u3 + u1 u2 + u1 + u2 u3 + u3 ) + (u1 u2 u3 + u1 u2 + u1 + u2 + u3 )x+ (u1 u2 u3 + u1 u2 + u1 u3 + u1 + u2 u3 + u2 + u3 )x2 + (u1 u2 u3 + u2 u3 )x3 + (u1 u2 u3 + u1 u3 + u1 + u2 )x4 [40, 20, 8]b−1 (u1 u2 u3 + u1 + u2 u3 + u3 ) + (u1 u2 u3 + u1 u3 + u1 )x + (u1 u2 + u2 u3 + u2 + u3 )x2 + (u1 u2 + u1 u3 + u1 )x3 + (u1 + u2 )x4 [40, 16, 10]∗−2

If we take n = 6 and f (x) = (u1 u2 u3 + u2 u3 + u2 ) + (u1 u2 u3 + u1 u2 + u1 u3 + u2 u3 + u2 + u3 )x + (u1 + u2 u3 + u3 )x2 + (u1 u2 u3 + u1 u2 + u3 )x3 + (u1 u2 u3 + u1 + u2 )x4 + (u2 + u3 )x5 , we get φ3 (< f >) to be a binary linear code with parameters [48, 20, 12]b If we take n = 7 and f (x) = (u1 u2 u3 + u1 u2 + u1 + u2 ) + (u1 u2 + u1 + u2 u3 + u3 )x + (u1 u2 u3 + u2 u3 + u2 + u3 )x2 + 12

(u1 u2 u3 + u1 u2 + u1 u3 + u1 + u2 + u3 )x3 + (u1 u2 u3 + u1 u2 + u1 + u2 u3 + u2 )x4 + (u1 u2 u3 + u1 u3 + u2 u3 + u2 + u3 )x5 + (u1 u2 u3 + u2 )x6 , we get φ3 (< f >) to be a binary linear code with parameters [56, 24, 12]b . If we take n = 7 and f (x) = u1 u2 u3 (x + x4 + x5 + x6 ), then we get φ3 (< f >) to be a binary linear code with parameters [56, 3, 32]∗ . If we take n = 8 and f (x) = (u1 + u2 + u3 + 1) + (u1 u2 u3 + u1 u3 + u2 + u3 + 1)x + (u1 u2 + u2 + u3 + 1)x2 +(u1 u2 u3 + u1 u2 + u1 + u2 u3 + u2 + 1)x3 + (u1 u2 + u1 u3 + u1 + u2 u3 + u2 + u3 + 1)x4 (u1 u2 u3 + u1 u2 + u1 + u2 u3 + u3 + 1)x5 + (u1 + u2 u3 + 1)x6 + (u1 u2 + u1 + u2 u3 + 1)x7 , we get φ3 (< f >) to be a binary linear code with parameters [64, 32, 12]b . If we take n = 8 again and f (x) = (u1 u2 u3 + u1 u3 + u1 + u2 u3 + u2 + u3 + 1) + (u1 u2 + u2 + 1)x + (u1 + u2 u3 + 1)x2 + (u1 u2 u3 + u1 + u2 u3 + u3 + 1)x3 + (u1 + u2 u3 + u3 + 1)x4 + (u1 u3 + u1 + 1)x5 + (u1 u2 + u1 + u2 u3 + u2 + 1)x6 + (u1 u2 u3 + u2 u3 + u2 + u3 + 1)x7 , we get φ3 (< f >) to be a binary linear code with parameters [64, 24, 16]b . If we take n = 10, and f (x) = (u1 u2 u3 + u1 u3 + u1 + u2 u3 + u2 + u3 ) + (u1 u2 + u1 + u2 u3 + u3 )x + (u1 u2 + u2 u3 )x2 + (u1 u3 + u1 + u2 u3 )x3 + (u1 u2 u3 + u1 u2 + u1 u3 + u1 + u2 u3 + u2 )x4 + (u2 + u3 )x5 + (u1 u2 u3 + u1 u3 +u1 + u2 )x6 + (u2 u3 + u2 + u3 )x7 + (u2 u3 + u3 )x8 + (u1 u2 u3 + u1 u3 + u1 + u2 + u3 )x9 , we get φ3 (< f >) to be a binary linear code with parameters [80, 36, 16]b−1 . Finally, if we take n = 12, and f (x) = (u1 u2 u3 +u3 +1)+(u1 u2 +u1 +u3 +1)x+(u1 u2 u3 +u1 u2 +u1 +u2 +u3 +1)x2 +(u1 u2 u3 + u1 + u2 u3 + u2 + u3 + 1)x3 + (u1 u2 u3 + u1 u2 + u1 u3 + u2 u3 + u2 + 1)x4 + (u1 u2 u3 + u1 u2 + u1 u3 + u2 u3 +u3 +1)x5 +(u2 +u3 +1)x6 +(u1 +u3 +1)x7 +(u1 u2 u3 +u1 u2 +u1 u3 +u1 +u2 u3 +u2 +1)x8 + (u1 u2 u3 + u1 u2 + u2 u3 + u3 + 1)x9 + (u1 u2 + u1 u3 + u1 + u2 u3 + u3 + 1)x10 + (u1 u2 u3 + u1 u3 + u1 + u2 u3 + u2 + 1)x11 , we get φ3 (< f >) to be a binary linear code with parameters [96, 48, 16]b . We want to finish this section by remarking that we were able to construct binary linear codes with parameters [80, 36, 16]b−1 , [80, 38, 14]b−2 , [64, 22, 16]b−2 and [80, 40, 14]b−2 as the φ4 images of one-generator cyclic codes over R4 . However we will not give the generators here. 13

5

The Images of Cyclic codes over Rk

5.1

Cyclic codes over Rk and the Nechaev Permutation

The Nechaev permutation was used to obtain results about the image of cyclic codes over Z4 in [20]. We can use the same idea to obtain some results about cyclic codes over Rk . n First, note that every vector in Rkn can be written uniquely as c + uk d, where c, d ∈ Rk−1 . n n Define Φk : Rk → Rk−1 as follows: Φk (c + ud) = (d, c + d).

(14)

Note that Φk is a linear distance-preserving map. We next recall the definition of the Necahev permutation from [20]: Definition 1. For odd n, let σ be the following permutation of {0, 1, 2 . . . , 2n − 1}: σ = (1, n + 1)(3, n + 3) · · · (2i + 1, n + 2i + 1) · · · (n − 2, 2n − 2). Then the Nechaev permutation on R2n is defined as: π(a0 , a1 , · · · , a2n−1 ) = (aσ(0) , aσ(1) , · · · , aσ(2n−1) ).

(15)

We then can give the following result as an analogue of Corollary 3.8 from [20]: Theorem 5.1. Let n be odd, and C be a cyclic code over Rk of length n. If D = Φk (C) is the image of C under Φk , then π(D) is a cyclic code of length 2n over Rk−1 . The proof can be done in an analogous way that was done in [20] and thus will be omitted here.

5.2

The Binary Images of Cyclic Codes over Rk under the Gray map

In [10] two Gray maps are described for the ring Rk which are shown to be conjugate. We shall recall the second Gray map and give a corresponding theorem about the image of cyclic codes under this map. View Rk as a vector space over F2 with basis {uA : A ⊆ {1, 2, . . . , k}}, and define the Gray map of each uA and then extend it linearly to all of Rk . Fix an ordering on the subsets of {1, 2, . . . , k}, that will be defined recursively as follows: {1, 2, . . . , k} = {1, 2, . . . , k − 1} ∪ {k}. k

We can now define the coordinate-wise Gray map. We denote this map by ψk : Rk → F22 and define it as follows: ψk (uA ) = (cB )B⊆{1,2,...,k} , 14

where

( cB =

1 ifB ⊆ A 0 otherwise.

We then extend ψk linearly to all of Rk and define the Lee weight of an element in Rk to be the Hamming weight of its image. We get a linear distance preserving map from Rkn to k F22 n . It follows immediately that wL (uA ) = 2|A| . (16) The map ψk was shown to be equivalent to φk in [10]. Viewing the action of ψk coordinatewise on vectors, i.e. ψk (a0 , a1 , · · · , an−1 ) = (ψk (a0 ), ψk (a2 ), · · · , ψk (an−1 )), we can prove the following lemma in an analogous way to [23]: Lemma 5.2. If τ is the cyclic shift operator on vectors of length n, then k

ψk ◦ τ = τ 2 ◦ ψk . But, now using this lemma it is easy to prove the following theorem: Theorem 5.3. Let C be a cyclic code of length n over the ring Rk . Then ψk (C) is a 2k -quasi-cyclic binary linear code of length 2k n.

6

Conclusion

We shall give a few paths for further research on cyclic codes over this family of rings. Udaya and Bonnecaze introduced a decoding algorithm for cyclic codes over F2 + uF2 in [19] which provided a strong motivation for studying cyclic codes over R1 = F2 + uF2 . One may consider an extension of this problem for cyclic codes over Rk as well. Given the importance of finding ideals in Rk , an interesting research problem would be to determine the number of ideals in Rk . We know the answer for k = 0, 1 and 2; but even for k = 3, it proves to be a difficult problem. As was seen in Section 3, the number of cyclic codes over Rk is closely related to the number of ideals of Rk . We believe the problem to be of interest algebraically as well. Another line of research could be finding better structural results for cyclic codes that are not one-generator cyclic codes. In particular, finding the size of a cyclic code and finding some bounds for the minimum distances of these codes seem to be challenging problems.

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References [1] T. Abualrub and R. Oehmke, On the generators of Z4 cyclic codes of length 2e , IEEE-IT, 49, No. 9, September 2003, 2126–2133. [2] T.Abualrub, I.Siap, Cyclic Codes over the rings Z2 + uZ2 and Z2 + uZ2 + u2 Z2 , Des. Codes Crypt. vol. 42, pp. 273–287, 2007. [3] T. Blackford, Cyclic codes over Z4 of oddly even length, Discrete Applied Mathematics, 128, 2003, 27–46. [4] A. Bonnecaze and P. Udaya, Cyclic Codes and Self-Dual Codes Over F2 + uF2 , IEEE Trans. Inform. Theory, vol. 45. pp. 1250-1255, 1999. [5] A.R. Calderbank and N.J.A. Sloane, Modular and p-adic cyclic codes, Designs, Codes and Cryptography, 6, 1995, 21–35. [6] H. Quang Dinh and S. Lopez-Perm´outh,Cyclic and Negacyclic Codes Over Finite Chain Rings, IEEE Trans. Inform. Theory, vol. 50, pp. 1728-1744, 2004. [7] S.T. Dougherty, P. Gaborit, M. Harada and P. Sol´e, Type II codes over F2 + uF2 , IEEE Trans. Infrom. Theory, vol. 45, pp.32–45, 1999. [8] S.T. Dougherty and S. Ling, Cyclic codes over Z4 of even length , Designs, Codes and Cryptography, 127-153, May 2006. [9] S.T. Dougherty and Y.H. Park, , On Modular Cyclic Codes , Finite Fields and their Applications Volume 13, Number 1, 31-57, 2007. [10] S.T. Dougherty, B. Yildiz and S. Karadeniz, Codes over Rk , Gray Maps and their Binary Images, in press, Finite Fields Appl. (2010), doi: 10.1016/j.ffa.2010.11.002 [11] W.C. Huffman, On the decomposition of self-dual codes over F2 + uF2 with an automorphism of odd prime order, Finite Fields and Applications, vol.13, pp.681–712, 2007. [12] W.C. Huffman, Codes and Groups, in: V.S.Pless, W.C. Huffman(Eds.), Handbook of Coding Theory, Elsevier, Amsterdam, 1998, pp. 1345–1440. [13] T.Hurley, Group Rings and Rings of Matrices, Inter. J. Pure & Appl. Math. no.3, pp. 319-335, 2006. [14] G.H. Norton and A. S˘al˘agean, On the structure of linear and cyclic codes over a finite chain ring, Appl. Alg. Engrg. Comm. & Comput., 10, 2000, 489–506.

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[15] P. Konmar and S. K. Lopez-Permouth, Cyclic Codes over Integers Modulo pm , Finite Fields and Applications, Vol 3, pp. 334–352, 2007. [16] S. Ling and P. Sol´e, On the algebraic structure of quasi-cyclic codes I: finite fields, IEEE-IT, 47, No. 7, November 2001, 2751–2760. [17] S. Ling and P. Sol´e, On the algebraic structure of quasi-cyclic codes II: chain rings, Des., Codes & Crypto., 30, 2003, 113–130. [18] V.S. Pless and Z. Qian, Cyclic codes and quadratic residue codes over Z4 , IEEE-IT, 42, No. 5, September 1996, 1594–1600. [19] P. Udaya and A. Bonnecaze, Decoding of Cyclic Codes over F2 + uF2 , IEEE Trans. Inform. Theory, vol. 45, pp. 2148–2157, 1999. [20] J.Wolfmann, Negacyclic and Cyclic Codes over Z4 , IEEE Trans. Inform. Theory, vol. 45, pp. 2527–2532, 1999. [21] J. Wood, Duality for modules over finite rings and applications to coding theory. Amer. J. Math., Volume 121, 555-575, 1999. [22] B.Yildiz, S.Karadeniz Linear Codes over F2 + uF2 + vF2 + uvF2 , Des. Codes Crypt., vol. 54, pp. 61–81, 2010. [23] B.Yildiz, S.Karadeniz Cyclic Codes over F2 + uF2 + vF2 + uvF2 , Des. Codes Crypt., vol.58, no.3, pp. 221–234 2011.

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