Cyclic codes over local Frobenius rings of order 16

We study cyclic codes over commutative local Frobenius rings of order 16 and give their binary images under a Gray map which is a generalization of the Gray maps on the rings of order 4. We prove that the binary images of cyclic codes are quasi-cyclic codes of index 4 and give examples of cyclic codes of various lengths constructed from these techniques including new optimal quasi-cyclic codes.


Introduction
Cyclic codes are one of the most interesting classes of linear codes both over fields and over rings. This is largely because of their rich algebraic structure. They were first studied by Prange in 1957 in [16] and since this early work, numerous papers have been written examining both their algebraic structure and their applications. The well known paper of Hammons et al. [10], sparked a great deal of interest in codes over rings and especially in rings which have an attached Gray map to the binary space. This present work draws from both sources as we study cyclic codes over a family of finite rings which admit a Gray map to the binary Hamming space.
The rings we consider can be either a chain ring or a non-chain ring. Numerous papers have studied cyclic codes over finite chain rings, see [2,5,12,15,17] for example. More recently codes over non-chain rings and their structures have been studied in [6][7][8]18].
In [13], Martinez and Szabo have classified the finite commutative local Frobenius rings of order 16. There are 12 such rings of which 7 are non-chain rings and 5 are chain rings. All of those rings are of order 16 with maximal ideal m, |m| = 8 and |Soc(R)| = 2. In [8], generating characters for each ring have been described.
Moreover, MacWilliams relations for symmetrized and complete weight enumerators have been given in a very concrete form for codes over these rings. In [7], a weight preserving Gray map has been defined from these 12 rings to the binary space and self-dual and formally self-dual codes over these rings have been studied.
In this paper, we study the structure of cyclic codes over finite commutative local Frobenius rings of order 16. Since the characterization of cyclic codes over finite chain rings was already studied in [5], we will describe the structure of the ideals of the ring R[x]/ x n − 1 where R is a local Frobenius non-chain ring of order 16. We prove that the binary images of cyclic codes are quasi-cyclic codes of index 4. Finally, we give some examples of various lengths including some new optimal codes.

Definitions and notations
2.1. Rings and codes. We begin by giving the necessary definitions for rings and for codes over rings. Throughout this paper all rings are assumed to be finite, commutative, Frobenius and have a multiplicative identity.
The Jacobson radical of a ring is defined to be the intersection of all maximal ideals of the ring. The socle of the ring is the sum of all the minimal one sided ideals of the ring. For a ring R, we denote the Jacobson radical as J(R) and the socle as Soc(R). If R is a finite ring then the following statements are equivalent: (1) R is a Frobenius ring; (2) as a left module, R ∼ = R R; (3) as a right module R ∼ = R R . A local ring is a ring with a unique maximal ideal. If the ideals of a local ring are linearly ordered then it is said to be a chain ring.
A code over a ring R of length n is a subset of R n . If, additionally, it is a submodule, then we say that the code is linear. We attach to the ambient space R n the standard inner-product, namely [v, w] = v i w i . The orthogonal is defined in the usual way as C ⊥ = {v | v ∈ R n , [v, w] = 0, ∀w ∈ C}. Notice that an ideal in a ring R is a code of length 1 and hence we can speak of a ⊥ if a is an ideal of R.

2.2.
Local Frobenius rings of order 16. We shall now describe the structure of the local Frobenius rings of order 16. Let R be a local Frobenius ring with maximal ideal m, with |m| = 8, such that |R| = 16. All local Frobenius rings of order 16 have two elements u and v such that m = u, v . There exists an element w in these rings with Soc(R) = w = {0, w}. The following diagram shows an ideal structure for any local Frobenius non-chain ring R of order 16.
Since the ring R is Frobenius we have that Soc(R) = m ⊥ . Any local Frobenius non-chain ring of order 16 has 5 non-trivial ideals. The maximal ideal m = u, v is of size 8, the socle is of size 2 and the remaining 3 ideals are of size 4.
The classification of local Frobenius rings of order 16 is given in [13]. In this paper, we will deal with rings of order 16 for which we can identify elements u, v, w which define the ideals in the ring and for which we can write each element of the ring uniquely in the form a + bu + cv + dw where a, b, c, d ∈ F 2 . There are 5 local Frobenius chain rings with this structure which are given as follows: 16 . For chain rings we can identify elements for u, v and w but the ideal structure is a bit different. For example, consider the ring Z 16 . Here u = 2, v = 4 and w = 8. This allows us to apply the theory developed in this case even though there are only 5 ideals as opposed to 7 ideals. There are chain rings of order 16 which do not have this property, for example, F 4 [x]/ x 2 , which has only 3 ideals. In these cases you cannot find elements u, v and w to correspond to the developed theory. There are 7 local Frobenius non-chain rings which are given as follows: There are also additional chain rings like F 4 [x]/ x 2 which have only one non-trivial ideal. We shall not deal with these rings here since we do not have the elements u, v and w at our disposal to construct the proper Gray map.
Given the ideal structure of the rings it is possible to write every element of the ring uniquely in the form a + bu + cv + dw where a, b, c, d ∈ F 2 . Note that this does not imply that the additive structure of the ring is Z 4 2 . In fact, the possibilities for the additive structure are Z 4 2 , Z 4 × Z 2 2 , Z 8 × Z 2 and Z 16 . The following lemma is immediate noting the units in a local ring are the elements not in the maximal ideal. The following Gray map was defined in [7]: This map is formed by taking the classical Gray map for local Frobenius rings of order 4 and applying it recursively. That is, for Z 4 and F 2 + uF 2 , we have the standard Gray map φ 1 , with where x = 2 for Z 4 and x = u for F 2 + uF 2 . Then applying this map recursively viewing a local Frobenius ring of order 16 as having x = u for φ 1 and x = v for φ 2 we have the map as defined above.
The following theorem appears in [7].
Theorem 2.2. If C is a linear code over R, where R is a local Frobenius ring of order 16, of length n, size 2 k and minimum Lee weight d, then φ(C) is a binary code of length 4n, size 2 k and minimum weight d.

Cyclic codes over local Frobenius rings of order 16
In this section, we study the structure of cyclic codes over local Frobenius nonchain rings of order 16. Let R be a local Frobenius non-chain ring of order 16. A cyclic shift on R n is the permutation τ defined by τ (c 0 , c 1 , . . . , c n−1 ) = (c n−1 , c 0 , . . . , c n−2 ).
A linear code over R is said to be a cyclic code if it is invariant under the cyclic shift. Any codeword c = (c 0 , c 1 , . . . , c n−1 ) in R n corresponds in the usual way to a polynomial c( If we consider our polynomials as elements of the ring then xc(x) modulo x n − 1 represents the cyclic shift of c. The next theorem follows in the usual way from this discussion. In order to understand cyclic codes over R we shall study the ideals in the ring Throughout we shall assume n to be odd so that gcd(n, char(R)) = 1.
3.1. The ring n . Let R be a local Frobenius ring of order 16. If R is a finite chain ring and n is relatively prime to the characteristic of R, then R[x]/ x n − 1 is a principal ideal ring ( [5]). If R is not a chain ring then the ring has non-principal ideals. This gives the following lemma. For the characterization of units and non-units in n , we will use a group ring representation of the ring n .
Let G = g be the cyclic group of order n and RG be the group ring where R is a local Frobenius ring of order 16 and n = R[x]/ x n − 1 . We can write n RG where we map a 0 + a 1 x + · · · + a n−1 x n−1 to a 0 + a 1 g + · · · + a n−1 g n−1 . By following the results given in [11], every element in RG corresponds a circulant matrix in the form: . . a n−1 a n−1 a 0 a 1 . . . a n−2 . . .
Therefore to every element in n , there is a corresponding circulant matrix in the form given above.
By the above description, we can characterize the units and zero divisors in n . Note that the determinant function det is a multiplicative map from matrices over a commutative ring R to the ring R. See [11] for a complete description. An element α ∈ n is a unit if and only if det(σ(α)) is a unit in R. We can write this statement as a corollary. Corollary 3.3. Let R be a local Frobenius non-chain ring of order 16. An element α = a 0 + a 1 x + · · · + a n−1 x n−1 is a unit in n if and only if det(σ(α)) is a unit in R. Hence, α is a non-unit in n if and only if det(σ(α)) ∈ m, where m is the maximal ideal of the ring R.

3.2.
The structure of cyclic codes over local Frobenius non-chain rings of order 16. Let R be a local Frobenius ring of order 16. We introduce a canonical map which is a homomorphism from R to F 2 and study cyclic codes by using this map. We study the case when the length n is odd.
Recall that any element in R has a unique representation as a + bu + cv + dw and that µ : where m is the maximal ideal of the ring R, is the canonical map defined by µ(a + bu + cv + dw) = a. We can extend the map µ so that it is defined from the ring of polynomials over R to the ring of polynomials over F 2 . It is easy to see that the kernel of µ is the maximal ideal m of the ring Two polynomials, f (x) and . If a ring R is local, then the polynomial ring R[x] may not be a unique factorization domain. However, regular polynomials have the unique factorization property, where a polynomial is said to be regular if it is not a zero divisor in that ring.
Lemma 3.4. Let R be a local Frobenius non-chain ring of order 16. Then 1 + Proof. Any element β in the maximal ideal of a local Frobenius non-chain ring of order 16 satisfies the property that β 4 = 0. Let α = 1+us 1 (x)+vs 2 (x)+ws 3 (x) and α = (1 + us 1 (x) + vs 2 (x) + ws 3 (x)) 3 . Then αα = 1 + u 4 s 4 ). It can be easily seen that the right side of the equality is equal to 1 for all local Frobenius non-chain rings of order 16, by noticing that each term other than 1 is either 0 because it is the fourth power of an element in the maximal ideal, which is always 0, or is 2 times something in the socle which has characteristic 2. Hence α is the inverse of α in R, so α is a unit.
are coprime if and only if µ(f (x)) and µ(g(x)) are coprime.
Conversely, if µ(f (x)) and µ(g(x)) are coprime then there exist polynomials f (x) and , we multiply both sides of the last equation by the inverse of 1 + us 1 (x) + vs 2 (x) + ws 3 (x). Then it follows that f (x) and g(x) are coprime. Definition 1. Let R be a local Frobenius ring of order 16. An ideal I of R[x] is a primary ideal provided ab ∈ I implies that either a ∈ I or b r ∈ I for some positive integer r.
In the usual way, we can define a basic irreducible polynomial in a local Frobenius non-chain ring of order 16 as follows.
Definition 2. Let R be a local Frobenius non-chain ring of order 16.
In [14], it is proven that any basic irreducible polynomial in a finite ring is primary. Therefore, we have the result in the case for the local Frobenius non-chain rings of order 16.  Proof. Since the polynomial x n − 1 is monic and its coefficients 1 and −1 are not elements of the maximal ideal of R, it is easy to see that x n − 1 is not a zero divisor in R[x], so it is a unit.
The previous lemma, gives the following theorem.
Theorem 3.8. The polynomial x n − 1 has a unique factorization into basic irreducible pairwise coprime polynomials as follows:  Proof. Let I be a non-zero ideal of R[x]/ f (x) and let g(x) + f (x) ∈ I for some and so gcd(µ(f (x)), µ(g(x))) = 1 or µ(f (x)). If gcd(µ(f (x)), µ(g(x))) = 1, by Lemma 3.5, f (x) and g(x) are coprime. Hence there exist polynomials a(x) and On the other hand, if gcd(µ(f (x)), µ(g(x))) = µ(f (x)), then µ(f (x)) divides µ(g(x)) and so there exist polynomials c(x) and s i (x), i ∈ {1, 2, 3}, in R[x] such that g(x) = f (x)c(x) + us 1 (x) + vs 2 (x) + ws 3 (x) and at least for one i ∈ {1, 2, 3.}, gcd(µ(f (x)), µ(s i (x))) = 1. Multiplying both sides of the last equation by the element 1 + f (x) , we get If the equality gcd(µ(f (x)), µ(s i (x))) = 1 is satisfied for i = 1 or both i = 1 and i = 3, then g(x) + f (x) ∈ u + f (x) . If the equality gcd(µ(f (x)), µ(s i (x))) = 1 is satisfied for i = 2 or both i = 2 and i = 3, then g( , v + f (x) , depending on whether the polynomials s i (x), i ∈ {1, 2, 3} are equal or not. Therefore I is a subset of one of the following ideals u + f ( From the last result, there exists a non-zero element h( , so gcd(µ(f (x)), µ(t 1 (x))) = 1. This contradiction implies that there exist k(x), l(x), m i (x), i ∈ {1, 2, 3}, such that t 1 (x)l(x)+f (x)k(x) = 1+um 1 (x)+vm 2 (x)+wm 3 (x). Multiplying both sides of the last equation by the inverse of 1 + um 1 (x) + vm 2 (x) + wm 3 (x), and using Lemma 3.4, we get where θ(x) is the inverse of 1 + um 1 (x) + vm 2 (x) + wm 3 (x). Multiply both sides of the last equation by u + f (x) , then we get Since I is an ideal, the right side of the last equation is in Consequently, I is equal to one of the following ideals: Recall that a code is cyclic over a local Frobenius non-chain ring of order 16 if and only if it is an ideal in the ring n = R[x]/ x n − 1 . The next theorem describes the ideals in n . Theorem 3.10. Let n be odd. Let x n − 1 = f 1 (x)f 2 (x) · · · f r (x) be the representation of x n − 1 as a product of basic irreducible pairwise coprime polynomials in R[x]. Let f i (x) denote the product of all f j (x) except f i (x). Then any ideal in n is a sum of the following ideals, Proof. Since the polynomials f i (x) are basic irreducible and pairwise coprime, it is easy to show that Applying the Chinese Remainder Theorem, we have Thus, if I is an ideal of n , then I = ⊕I i , where I i is an ideal of the ring R[x]/ f i (x) for i = 1, . . . , r. By Lemma 3.9, I i is equal to one of the following ideals: 0 , , then it corresponds to the ideal f i (x) + x n − 1 in the ring n . If I i = α + f (x) , then it corresponds to the ideal α f i (x) + x n − 1 , α ∈ {u, v, u + v, w}.
, v + f (x) , then it corresponds to the ideal u f i (x) + x n − 1 , v f i (x) + x n − 1 . Hence I is a sum of the following ideals, The following corollary deals with the number of cyclic codes.
Corollary 3.11. Let R be a local Frobenius ring of order 16. If x n − 1 = f 1 (x)f 2 (x) · · · f r (x) then the number of cyclic codes over R of length n, where R is a chain ring, is The number of cyclic codes over R, where R is a non-chain ring, is 7 r .
Proof. Let R be a local Frobenius ring of order 16. If R is a chain ring then it has 3 ideals if R/m is F 4 and 5 ideals if R/m is F 2 . If R is a non-chain ring then it has 7 ideals. By the previous theorem the result follows.
As an example of the chain ring case, the ring F 4 [x]/ x 2 has 3 ideals where as Z 16 has 5 ideals.
For the rest of the paper, we choose to write the ideals of n = R[x]/ x n − 1 dropping the residue part x n − 1 . For simplicity we often write the polynomial f (x) more simply as f . Let be the unique factorization of x n − 1 into a product of monic basic irreducible pairwise coprime polynomials. Define the following polynomials for integers k i with k i = r: The next theorem discusses the generators of a cyclic code over a local Frobenius non-chain ring of order 16.
The cardinality of the cyclic code given by (2) is where K is a combination of the generators of u F 2 , v F 2 . Hence, C is generated by uf 1 + vf 2 f 3 .
As with cyclic codes over R, where R is a local Frobenius ring of order 16, we can find generator polynomials of the dual codes. Before studying generators of the dual code, we need to state the following well-known lemma [3]. The next theorem discusses the structure of the generators of the dual code.
Theorem 3.14. Let F 1 , K, u F 3 , v F 4 , (u + v) F 5 , w F 6 be a cyclic code over a local Frobenius non-chain ring of order 16 with |C| = 2 s , s = 4(n − deg F 1 ) where K is a combination of the generators of u F 2 , v F 2 and x n − 1 = F 0 · · · F 6 . Then . . , 6, and M * is the reciprocal polynomial of M , which is a combination of the generators of the ideal u F 6 , v F 6 and |C ⊥ | = 2 4degF0+3degF6+2(degF3+degF4+degF5)+degF2 .
Proof. We assume that where M * is the reciprocal polynomial of M , which is a combination of generators of the ideal u F 6 , v F 6 . First observe that 3.3. One generator cyclic codes. In this subsection, we describe one generator cyclic codes over a local Frobenius non-chain ring R. Let p(x) = a 0 + a 1 x + · · · + a n−1 x n−1 be a polynomial in R[x] and C be the cyclic code of length n generated by p(x) in n . We will assume that p(x) is not a unit, otherwise the cyclic code generated by p(x) would be the trivial code R n . Hence we assume that p(x) is a non-unit in n .
Recall that the map µ : R → F 2 is the canonical map which was defined previously with µ(a + bu + cv + dw) = a. This means that µ reduces the elements of R modulo u, v. The function µ maps every unit in R to 1, and every non-unit to 0. Since µ is a ring homomorphism, we can write det(σ(µ(p(x)))) = µ(det(σ(p(x)))).
It can be seen easily that σ(µ(p(x))) is a matrix with entries 0 or 1. Then, any element p(x) is a non-unit in n if and only if σ(µ(p(x))) is a singular matrix. Hence, p(x) generates a non-trivial cyclic code C over R if and only if σ(µ(p(x))) is a singular matrix. Hence the binary cyclic code µ(C) generated by µ(p(x)) is nontrivial. Then, since µ(p(x)) is a cyclic code over F 2 , it is a non-unit in F 2 [x]/ x n −1 . We know that any polynomial µ(p(x)) in F 2 [x] is a unit in F 2 [x]/ x n − 1 if and only if gcd(µ(p(x)), x n − 1) = 1. Hence we have proven the following theorem.

The binary images of cyclic codes under the Gray map
In this section we study the binary images of cyclic codes under the Gray map. Recall that τ is the cyclic shift as defined in Section 3. A code C is said to be an r-quasi-cyclic code if it is invariant under τ r . Notice that we are not assuming that the code is linear, but rather that is invariant under τ r . We call r the index of the quasi-cyclic code.
Recall that each element of a local Frobenius non-chain ring R of order 16 can be written uniquely as a + bu + cv + dw.
It can be easily proven that φ is linear for these rings.
Proof. Take an element c ∈ R n , then On the other hand we can write Hence, we get the proof from (4) and (5).
We have the following theorem. Proof. Since C is a cyclic code, we have τ (C) = C where τ is the cyclic shift which is defined above. Apply the map Φ to both sides Φ(τ (C)) = Φ(C).
We shall give some concrete examples of cyclic codes in this setting and summarize our computational results, including optimal codes, in Tables 1,2, 3 and 4.
The automorphism group of C has order 2 2 × 23.
The binary image of the code is a 28, 2 9 , 10 * 2 code that has the same minimum distance as the corresponding optimal binary linear code.
For the following tables a polynomial f (x) = a n−1 x n−1 +a n−2 x n−2 +· · ·+a 2 x 2 + a 1 x + a 0 is abbreviated as (a 0 , a 1 , a 2 , . . . , a n−2 , a n−1 ). Throughout the tables, the symbols * and b denote that the corresponding binary code is optimal or the best known code with respect to the online database of linear codes [9].