On negacyclic codes over the ring ℤ_p + uℤ_p + . . . + u^{k + 1} ℤ_p

1 Introduction

In the middle last of the century, Shannon, Golay and Hamming discussed codes over finite field. Recently, codes over chain ring have been considered as the finite ring can be related to codes over finite fields via the Gray map. Coding theory, at its origin, is concerned with effective communication of information. Specifically, to detect and correct transmission errors techniques were developed. Coding theory was primarily studied for its application to telephone communication, elementary computing machines and computer to computer communication.

Negacyclic code played a very important role in the area of error correcting codes due to their rich algebraic structure. Negacyclic code has a lot of applications in consumer electronics, data transmission technologies, broadcast systems and computer applications as it has efficient encoding and decoding algorithms. In 1968, Berklamp investigated negacyclic codes over odd prime fields GF(p) and designed a decoding algorithm that corrects up to t ≤ ⎣ (p − 1)/2⎦ in [1]. In [2], Wolfmann discussed some interesting results about negacyclic codes of odd length over ℤ₄, and proposed questions about such codes when the length is even. In [3], Blackford classified all negacyclic codes of even length using a transform approach. This has motivated authors to study the negacyclic codes over the ring ℤ_p [u]/〈 u^k 〉. In particular, cyclic and negacyclic codes over finite rings have received much attention in series of paper [4–5].

Dinh and Lopez-Permouth in [16] have considered the ring ℤ_2^m(m ≥ 2) and studied the negacyclic codes of length 2^t(t ≥ 1) over that. They have provided complete list of negacyclic codes of length 2^t Over ℤ_2m. Dinh in [17], determined the structure of negacyclic codes of length 2^s over the Galois ring GR(2^a, m), as well as that of their duals, and studied the Hamming distance of negacyclic codes of length 2^s over the Galois ring GR(2^a, m).

Let R_{k, p} = ℤ_p + uℤ_p + … + u^k-1ℤ_p, where p is a prime number and u^k = 0. Note that the ring R_{k, p} can also be viewed as the quotient ring ℤ_p[u]/〈u^k〉. Let C be a negacyclic code over the ring R_{k, p} = ℤ_p + uℤ_p + … + u^{k – 1}ℤ_p, u^k = 0. The line of arguments we have used to find a set of generators and a minimal spanning set of a code C are somewhat similar to those discussed in [4]. The idea to find a set of generators is as follows. We view the negacyclic code C as an ideal in R_{k, p,p^s} = R_{k, p}/〈 x^ps + 1 〉. Then we define the isomorphism map from μ : R_{k, p}[x]/ 〈 x^ps − 1〉 → R_{k, p}[x]/ 〈 x^ps + 1〉 and discuss that I is a cyclic code of length p^s over R_{k, p} if and only if μ(I) is a negacyclic code of length p^s over R_{k, p}.

This paper is organized as follows. In second section, we present some basic background. In Section 3, we determine a set of generators for the negacyclic codes C over the ring R_{k, p} = ℤ_p + uℤ_p + … + u^{k ℒ 1}ℤ_p, u^k = 0. We discuss a minimal spanning set for these codes and find the rank in section 4. In Section 5, we study the minimum distance of these codes. We discuss some examples of different lengths of these codes in Section 6. Section 7 concludes the paper.

2 Preliminaries

Let R be a ring. A linear code of length n over R is a R submodule of Rⁿ. A linear code C of length n over R is λ-constacyclic if for some unit λ ∈ R, the code is invariant under the automorphism ν. That is ν(c_{n − 1}, c₁, …, c₀) = (λc₀, c₁, …, c_{n − 1}) . We can consider a negacyclic codes C(λ = − 1) of length n over R as an ideal in R[x]/〈xⁿ + 1 〉 via the following correspondence

A ring R is called local ring if R has a unique maximal right (left) ideal. A ring is called a chain ring if the set of all right (left) ideals of R is a chain under set-theoretic inclusion. If R is both a right and a left chain ring, we simply call R a chain ring.

For the class of finite commutative rings, We have the following equivalent conditions.

Let R be a finite commutative local ring with maximal ideal M. We define the residue field R = R/M. Let μ : R[x] → R[x] denote the natural ring homomorphism that maps r ↦ r + M and the variable x to x. We define the degree of the polynomial f(x) ∈ R[x] as the degree of the polynomial μ(f(x)) in R[x], i.e., deg(f(x)) = deg(μ(f(x)). A polynomial f(x) ∈ R[x] is called regular if it is not a zero divisor.

Proof: (i) ⇒ (ii) Let μ(f(x)) ≠ 0 ⇒ ∃ an a_i for some i, 0 ≤ i ≤ n such that a_i ∉ M. ⇒ a_i is a unit element (Since all element out side of the M are unit).

(ii) ⇒ (iii) Let a_i is a unit for some i, 0 ≤ i ≤ n ⇒ 〈a_i〉 = R ⇒ 〈a₀, a₁, …, a_n〉 = R.

(iii) ⇒ (i) Let 〈a₀, a₁, …, a_n〉 = R. We claim that at least one of a_i is a unit for some i, 0 ≤ i ≤ n. If not then all a_i ∈ ⇒ 〈 a₀, a₁, …, a_n〉 ⊆ M ⊂ R. A contradiction. So ∃ an a_i ∉ M for some i, 0 ≤ i ≤ n ⇒ μ(f(x)) ≠ 0.

The following version of the division algorithm holds true for polynomials over finite commutative local rings.

Let R_{k, p} = ℤ_p + uℤ_p + … + u^{k − 1}ℤ_p, u^k = 0. It is easy to see that the ring R_{k, p} is a finite chain ring with unique maximal ideal 〈u〉. Let g(x) be a non zero polynomial in ℤ_p[x]. It is also easy to see that the polynomial g(x) + up₁(x) + u²p₂(x) + … + u^{k − 1}p_{k − 1}(x) ∈ R_{k, p}[x] is regular. Throughout the paper, we repeatedly make use for the polynomial g(x) + up₁(x) + u²p₂(x) + … + u^{k −1}p_{k − 1}(x) ∈ R_{k, p}[x]. Note that deg(g(x) + up₁(x) + u²p₂(x) + … + u^{k − 1}p_{k − 1}(x)) = deg(g(x)).

3 A generator for negacyclic codes of length p^s over the ring R_{k, p}

Let p be a odd prime number. Let R_{k, p} = ℤ_p + uℤ_p + ⋯ + u^{k − 1}ℤ_p, u^k = 0.

Throughout this paper we use the isomorphism μ and restriction of μ defined in Proposition [3.1] and Remark [3.1].

Proof: The proof is obvious since the map μ is a ring isomorphism.

Let p be a odd prime number. Let R_{k, p} = ℤ_p + uℤ_p + ⋯ + u^{k − 1}ℤ_p, u^k = 0. and R_{k, p, p^s} = R_{k, p}[x]/〈 x^ps + 1〉. Let C_k be a negacyclic code of length p^s over R_{k, p}. From Corollary [3.1], we know that for the negacyclic code C_k there exist a cyclic code, say A over the same ring and of same length. We know the structure of a cyclic code over the ring R_{k, p} from [4]. Since p^s is not relatively prime to p, then we know that cyclic code of length p^s over R_{k, p} is A = 〈g(x) + up₁(x) + u²p₂(x) + ⋯ + u^k−1p_k−1(x), ua₁(x) + u²q₁(x) + ⋯ + u^k−1q_k−2(x), u²a₂(x) + u³l₁(x) + ⋯ + u^{k − 1}l_k−3(x), ⋯, u^{k − 2}a_{k− 2}(x) + u^k−1t₁(x), u^k−1a_k−1(x)〉 with a_k−1(x)|a_k−2(x)| ⋯ | a₂(x)|a₁(x)|g(x)|(x^ps−1) mod p, ak−2(x)|p1(x)(xps−1g(x)),⋯,ak−1|t1(x)(xps−1ak−2(x))⋯,ak−1|pk−1×(xps−1g(x))⋯(xps−1ak−2(x)) Moreover, deg p_k−1(x) 〈dega_k−1(x), ⋯, deg t₁(x) 〈a_k−1(x), ⋯, and deg p₁(x)〈 dega_k−2(x). Now the polynomials g(x) + up₁(x) + u²p₂(x) + ⋯ + u^k−1p_{k − 1}(x), ua₁(x) + u²q₁(x) + ⋯ + u^k−1q_k−2(x),⋯,u^k−2a_k−2(x) + u^k−1t₁(x), u^k−1a_k−1(x) ∈ R_{k, p}[x]/〈 x^ps − 1〉 Therefore from the definition of μ from Proposition [3.1], we get μ(g(x) + up₁(x) + u²p₂(x) + ⋯ + u^k−1p_k−1(x)) = g(-x) + up₁(-x) + u²p₂(-x) + ⋯ + u^k−1p_k−1(-x) = g₁(x) + up₁₁(x) + u²p₁₂(x) + ⋯ + u^k−1p_1(k−1)(x), μ(ua₁(x) + u²q₁(x) + ⋯ + u^k−1q_k−2(x)) = ua₁(-x) + u²q₁(-x) + ⋯ + u^k−1q_k−2(−x) = ua₁₁(x) + u²q₁₁(x) + ⋯ + u^k−1q_1(k−2)(x), ⋯, μ(u^k−2a_k−2(x) + u^k−1t₁(x)) = u^k−2a_k−2(−x) + u^k−1t₁(−x) = u^k−2a_1(k−2)(x) + u^k−1t₁₁(x), μ(u^k−1a_k−1(x)) = u^k−1a_k−1(−x) = u^k−1a_1(k−1)(x) and μ(x^ps-1) = −(x^ps + 1), where g(−x) = g₁(x), p₁(−x) = p₁₁(x), p₂(−x) = p₁₂(x),⋯, a_k−2(−x) = a_1(k−2)(x), t₁(−x) = t₁₁(x), a_k−1(−x) = a_1(k−1)(x). Again μ(A) = C_k. Therefore the code C can be written as C_k =〈 g₁(x) + up₁₁(x) + u²p₁₂(x) + ⋯ + u^k−1p_1(k-1)(x), u a₁₁(x) + u²q₁₁(x) + ⋯ + u^k−1q_1(k−2)(x), u²a₁₂(x) + u³l₁₁(x) + ⋯ + u^k−1l_1(k−3)(x), …, u^k−2a_1(k−2)(x) + u^k−1t_{11(x), u^k−1a1(k−1)(x)〉 with a1(k−1)}(x)|a_1(k−2)(x)| ⋯ | a₁₂(x)|a₁₁(x)|g₁(x)|(x^ps+1) mod p, a1(k−2)(x)|p11(x)(xps+1g(x)),⋯,a1(k−1)|t11(x)(xps+1a1(k−2)(x)),⋯,a1(k−1)|p1(k−1)×(xps+1g1(x))⋯(xps+1a1(k−2)(x)).

Proof: We have u (g₁(x) + u p₁₁(x)) = ug₁(x) and g(x) = a₁₁(x). It is clear that C₂ ⊂ 〈g₁(x) + up₁₁(x)〉. Hence, C₂ = 〈 g₁(x) + up₁₁(x)〉. By the division algorithm, we have x^ps − 1 = (g₁(x) + up₁₁(x)) q(x) + r(x), where r(x) = 0 or deg r(x) 〈 r. This implies that r(x) = (x^{ps − 1) − (g₁(x) + up₁₁(x)) q(x). This gives, r(x) ∈ C₂. Thus, we have r(x) = 0 and hence (g₁(x) + up₁₁(x))|(x^ps − 1) in R₂.

4 Ranks and minimal spanning sets

If negacyclic code C over the finite chain ring R are free R-submodules, then the basis of C over R is called the minimal generating set of C, and the number of element include in the minimal generating set is called the rank of the code C, denoted the rank(C).

Proof: (1) Suppose x^ps+1 = (g₁(x) + u p₁₁(x))(h₁(x) + u h₁₁(x)) over R₂. Let c(x) ∈ C₂ = 〈 g₁(x) + u p₁₁(x)〉, then c(x) = (g₁(x) + up₁₁(x))f(x) for some polynomial f(x). If deg f(x) ≤ p^s−r−1, then c(x) can be written as linear combinations of elements of B₁. Otherwise by the division algorithm there exist polynomials q(x) and r(x) such that f(x)=(xps+1g1(x)+up11(x))q(x)+r(x) where r(x) = 0 or deg r(x) ≤ p^s−r−1. This gives,

Since deg r(x) ≤ p^s−r−1, this shows that B₁ spans C₂. Now we only need to show that B₁ is linearly independent. Let g₁(x) = g₁₀ + g₁₁x + ⋯ + g_1rx^r and p₁₁(x) = p₁₁₀ + p₁₁₁x + ⋯ + p_11lx^l, g₀ ∈ ℤp×, g_1i, p_11(i−1) ∈ ℤ_p, i ≥ 1. Suppose (g₁(x) + u p₁₁(x))c₀ + x(g₁(x) + u p₁₁(x))c₁ + ⋯ + x^ps−r−1(g₁(x) + u p₁₁(x))c_{p^s−r−1 = 0. By comparing the coefficients in the above equation, we get

Since (g₁₀ + up₁₁₀) is unit, we get c₀ = 0. Thus,

Again comparing the coefficients, we get

As above, this gives c₁ = 0. Continuing in this way we get that c_i = 0 for all i = 0, 1. …, n−r−1. Therefore, the set B₁ is linearly independent and hence a basis for C₂.

(2) If C₂ = 〈g₁(x) + up₁₁(x), ua₁₁(x) 〉 with deg g₁(x) = r and deg a₁₁(x) = t. The lowest degree polynomial in C₂ is ua₁₁(x). It is suffices to show that B₂ spans B = {g₁(x) + up₁₁(x), x(g₁(x) + up₁₁(x)), · · ·, x^ps−r−1(g₁(x) + up₁₁(x)), ua₁₁(x), xua₁₁(x), · · ·, x^ps−t−1ua₁₁(x)}. We first show that ux^r−ta₁₁(x) ∈ span(B₂). Let the leading coefficients of x_r−ta₁₁(x) be a₀ and of g₁(x) + up₁₁(x) be g₁₀. There exists a constant c₀ ∈ ℤ_p such that a₁₀ = c₀g₁₀. Then we have

where um(x) is a polynomial in C₂ of degree less than r. Since C₂ = < g₁(x) + up₁₁(x), ua₁₁(x)>, any polynomial in C₂ must have degree greater or equal to deg a₁₁(x) = t. Hence, t ≤ deg m(x) < r and

Thus, ux^r−ta₁₁ ∈ span(B₂). Inductively, we can show that ux^r−t+1a(x), ⋯ ux^ps−t−1a(x) ∈ span(B₂). Hence, B₂ is a generating set. As in (1), by comparing the coefficients we can see that B₂ is linearly independent. Therefore, B₂ is a minimal spanning set and |C₂| = p^2ps−r−t. Following the same process as in the above theorem, we can find the rank and the minimal spanning set of any cyclic code over the ring R_{k, p}, k ≥ 1.

5 Minimum Distance

Since p^s is not relatively prime to p. Let C₂ = 〈 g₁(x) + up₁₁(x), ua₁₁(x)〉 be a negacyclic code of length p^s over R₂ = ℤ_p + u ℤ_p, u² = 0. We define C_2,u = k(x) ∈ R_2,n : u k(x) ∈ C₂. It is easy to see that C_2,u is a negacyclic code over ℤ_p. Let C_k be a negacyclic code of length p^s over R_{k, p} = ℤ_p + u ℤ_p + ⋯ + u^k−1ℤ_p, u^k = 0. We define C_{k, u^k−1} = k(x) ∈ R_{k, n} : u^k−1k(x) ∈ C_k. Again it is easy to see that C_{k, u^k−1} is a negacyclic code over ℤ_p.

Proof: We have u^k−1a_1(k−1)(x) ∈ C_k, thus 〈a_1(k−1)(x)〉 ⊆ C_{k, u^k−1}. If b(x) ∈ C_{k, u^k−1}, then u^k−1b(x) ∈ C_k and hence there exist polynomials b₁(x), ⋯, b_k(x) ∈ ℤ_p[X] such that u^k−1b(x) = b₁(x)u^k−1g₁(x) + b₂(x)u^k−1a₁₁(x) + b₂(x)u^k−1a₁₂(x)+ ⋯ + b_k(x)u^k−1a_1(k−1)(x). Since a_1(k−1)(x)|a_1(k−2)(x)|⋯ |a₁₂(x)| a₁₁(x)|g₁(x), we have u^k−1b(x) = m(x)u^k−1a_1(k−1)(x) for some polynomial m(x) ∈ ℤ_p[x]. So, C_{k, u^k−1} ∈ 〈 a_1(k−1)(x)〉, and hence C_{k, u^k−1} = 〈 a_1(k−1)(x)〉. Let m(x) = m₀(x) + um₁(x) + ⋯ + u^k−1m_k−1(x) ∈ C_k, where m₀(x), m₁(x), ⋯, m_k−1(x) ∈ ℤ_p[x]. We have u^k−1m(x) = u^k−1m₀(x), w_H(u^k−1m(x)) ≤ w_H(m(x)) and u^k−1C_k is subcode of C_k with w_H(u^k−1C_k) ≤ w_H(C_k). Therefore, it is sufficient to focus on the subcode u^k−1C_k in order to prove the theorem. Since u^k−1C_k = 〈 u^k−1a_1(k−1)(x)〉, we get w_H(C_k) = w_H(C_k, u^k−1).

Proof: For c ∈ C, we have c = (x^ps−1 - 1)^bh(x)m(x) for some m(x)∈Rk,p[x](xps−1). Since h(x) generates a negacyclic code of length p^s−1, we have w(c) = w((x^ps−1 − 1)^bh(x)m(x)) = w(x^ps−1bh(x)m(x)) + w(^bC₁x^{ps−1(b−1)}h(x)m(x)) + ⋯ + w(^bC_b−1x^ps−1h(x)m(x)) + w(h(x)m(x)). Thus, d(C) = (b + 1)d.

6 Examples

In this section, we give some examples of negacyclic codes of different lengths over the ring R_{k, p}.