Symbol-Pair Distance of Repeated-Root Constacyclic Codes of Prime Power Lengths over F p m [ u ] / u

: Let p be a prime, s , m be positive integers, γ be a nonzero element of the ﬁnite ﬁeld F p m , and let R = F p m [ u ] / (cid:104) u 3 (cid:105) be the ﬁnite commutative chain ring. In this paper, the symbol-pair distances of all γ -constacyclic codes of length p s over R are completely determined.


Introduction
Initially, in information theory, the message communicated in a noisy channel was divided into information units, which were called individual symbols. The research on the process of writing and reading is often presumed to be performed on individual symbols. With the development of high-density data storage technologies, symbol-pair codes are proposed to protect efficiently against a certain number of pair-errors. In [1,2], Cassuto and Blaum established a new coding framework for channels whose outputs are overlapping pairs of symbols. In 2011, by using algebraic methods, Cassuto and Litsyn [3] constructed cyclic symbol-pair codes. Applying Discrete Fourier Transform for coefficients of codeword polynomials c(x) ∈ F q [x]/ x n − 1 and BCH bounds, Cassuto and Litsyn proved that for a cyclic code with dimensions greater than 1 and Hamming distance d H , the corresponding symbol-pair distance is at least d H + 2 [3] [Th. 10]. In particular, Kai et al. [4] extended the result of Cassuto and Litsyn [3] [Th. 10] for the case of simple-root constacyclic codes. Many researchers have scrutinized symbol-pair distances over constacyclic codes since then in [5][6][7][8][9] over many years.
Constacyclic codes are the pivotal and profound part of linear codes. It includes as a subclass the important class of cyclic codes, which form the most important and well studied class of error-correcting codes. This family of codes is thus interesting for both theoretical and practical reasons. Repeated-root constacyclic codes were first initiated in the most generality by Castagnoli in [10] and Van Lint in [11]. They established that the repeated-root constacyclic codes have a sequential structure, which motivated the researchers to further study these codes.
For any a ≥ 2, let R be the ring F p m [u]/ u a = F p m + uF p m + · · · + u a−1 F p m (u a = 0). The ring R has been widely used as alphabets in certain constacyclic codes (see, for instance ( [12][13][14]).
When a = 3, in 2015, the authors of [24] determined the structure of (δ + αu 2 )constacyclic codes of length p s over F p m [u]/ u 3 = F p m + uF p m + u 2 F p m . DNA cyclic codes over F 2 + uF 2 + u 2 F 2 were studied in [25]. In [26], Laaouine et al. obtained the structure of all γ-constacyclic codes of length p s over F p m + uF p m + u 2 F p m by classifying them into eight types, where γ is a nonzero element of F p m . The Hamming distances of γ-constacyclic codes of length p s over R have been computed by Dinh et al. [27]. Symbol-pair distances of γ-constacyclic codes have remained open. Motivated by that, we solved this problem in this paper.
The organization of this paper is as follows. Some preliminary results are discussed in Section 2. In Section 3, the symbol-pair minimum distances of γ-constacyclic codes of length p s are established over the ring R. Section 4 contains some examples for different values of p and s. We conclude the paper in Section 5.

Some Preliminaries
For a finite ring R, consider the set R n of n-tuples of elements from R as a module over R in the usual way. A code C of length n over R is a nonempty subset of R n and the ring R is referred to as the alphabet of C. In addition, C is called a linear code if C is an R-submodule of R n .
Let λ be an invertible element of R. The λ-constacyclic shift τ λ on R n is defined as and a code C is said to be λ-constacyclic if τ λ (C) = C, i.e., if C is closed under the λconstacyclic shift τ λ . In case λ = 1, those λ-constacyclic codes are called cyclic codes, and when λ = −1, such λ-constacyclic codes are called negacyclic codes. Each codeword c = (c 0 , c 1 , . . . , c n−1 ) ∈ C is customarily identified with its polynomial representation c(x) = c 0 + c 1 x + · · · + c n−1 x n−1 , and the code C is in turn identified with the set of all polynomial representations of its codewords. Then in the ring R[x]/ x n − λ , xc(x) corresponds to a λ-constacyclic shift of c(x). From that, the following fact follows at once (cf. [28,29]).

Proposition 1.
A linear code C of length n is λ-constacyclic over R if and only if C is an ideal of R[x]/ x n − λ .
In 2010, Cassuto and Blaum [1] gave the definition of the symbol-pair distance as the Hamming distance over the alphabet (Σ, Σ). Given x = (x 0 , x 1 , . . . , x n−1 ), y = (y 0 , y 1 , . . . , y n−1 ), the symbol-pair distance between x and y is defined as The symbol-pair distance of a symbol-pair code C is defined as The symbol-pair weight of a vector x is defined as the Hamming weight of its symbolpair vector π(x): If the code C is linear, its symbol-pair distance is equal to the minimum symbol-pair weight of nonzero codewords of C: Throughout this paper, let p be a prime, s, m be positive integers, F p m be the finite field of order p m , and let R = F p m [u]/ u 3 be the finite commutative chain ring with unity.
By applying Proposition 1, all γ-constacyclic codes of length p s over R are precisely the ideals in the ring where γ is a nonzero element of F p m . In [26], Laaouine et al. classified all γ-constacyclic codes of length p s over R.
Theorem 1 (cf. [26]). The ring R γ is a local finite non chain ring with maximal ideal u, x − γ 0 , where γ 0 ∈ F p m such that γ p s 0 = γ. The γ-constacyclic codes of length p s over R, that is, ideals of the ring R γ , are Type 2 (C 2 ) : Type 4 (C 4 ) : is a unit in R γ or 0, and L is same as in Type 3.
Type 5 (C 5 ) : Here U being the smallest integer such that u(x − γ 0 ) U + u 2 g(x) ∈ C 5 , for some g(x) ∈ R γ and V is the smallest integer satisfying u 2 (x − γ 0 ) V ∈ C 5 .
Type 6 (C 6 ) : Here U, V are same as in Type 5.
Type 7 (C 7 ) : Here W is the smallest integer such that u 2 (x − γ 0 ) W ∈ C 7 and U is same as in Type 5.
Type 8 (C 8 ) : , U as in Type 5 and W is same as in Type 7.
Proposition 2 (cf. [26]). We have Theorem 2 (cf. [26]). Let C be a γ-constacyclic codes of length p s over R. Then following the same notations as in Theorem 1, we have the following results:

Symbol-Pair Distance
In this section, we shall determine symbol-pair distances of all γ-constacyclic codes of length p s over R. To do this, we need the following theorem.
Theorem 3 (cf. [6]). Let C be a γ-constacyclic code of length p s over F p m . Then C = (x − γ 0 ) κ , for κ ∈ {0, 1, . . . , p s }, and its symbol-pair distance d sp (C) is completely determined by: Note that F p m is a subring of R, for a code C over R, we denote d sp (C F ) as the symbol-pair distance of C| F p m . Now, we compute the symbol-pair distance for each type of γ-constacyclic codes of length p s over R one by one.
For a code C 2 = u 2 (x − γ 0 ) τ of Type 2, 0 ≤ τ ≤ p s − 1, the codewords of C 2 are exactly same as the codewords of the γ-constacyclic codes ( , which are given in Theorem 3. be a γ-constacyclic codes of length p s over R of Type 2 (as classified in Theorem 1). Then the symbol-pair distance d sp (C 2 ) of the code C 2 is given by Now, we are going to determine the symbol-pair distances of those codes for the remaining cases (Type 3, 4, 5, 6, 7 and 8). To do this, we first observe that wt sp (a(x)) ≥ wt sp (ua(x)), where a(x) ∈ R γ . The symbol-pair distance of Type 3 γ-constacyclic codes can be calculated as follows: be a γ-constacyclic codes of length p s over R of Type 3 (as classified in Theorem 1). Then the symbol-pair distance d sp (C 3 ) of C 3 is given by Thus, By (1), we obtain that From this, we obtain wt sp (c(x)) ≥ d sp ( (x − γ 0 ) L F ) for each c(x) nonzero element of C 3 . This implies that On the other hand, we have that which implies that Now by (2) and (3), we obtain Now by applying Theorem 3, we obtain the desired result. Now, we determine the symbol-pair distance of Type 4 γ-constacyclic codes.
) ω be a γ-constacyclic code of length p s over R of Type 4 (as classified in Theorem 1). Then the symbol-pair distance d sp (C 4 ) of C 4 is given by Proof. First of all, since u 2 (x − γ 0 ) ω ∈ C 4 , it follows that Now by applying Theorem 3, we obtain the desired result.
Next, we calculate the symbol-pair distance of Type 5 γ-constacyclic codes as follows: codes of length p s over R of Type 5 (as classified in Theorem 1). Then the symbol-pair distance d sp (C 5 ) of C 5 is given by Thus, By (1), we see that . On the other hand we have that . Now by applying Theorem 3, we obtain the desired result. The symbol-pair distance of Type 6 γ-constacyclic codes can be established as follows: γ-constacyclic codes of length p s over R of Type 6 (as classified in Theorem 1). Then the symbol-pair distance d sp (C 6 ) of C 6 is given by Proof. First of all, since u 2 (x − γ 0 ) c ∈ C 6 , it follows that Now, consider an arbitrary polynomial c(x) ∈ C 6 \ u 2 (x − γ 0 ) c . Thus, by (1), we obtain that Now by applying Theorem 3, we obtain the desired result. Now, we determine the symbol-pair distance of Type 7 γ-constacyclic codes.
be a γ-constacyclic codes of length p s over R of Type 7 (as classified in Theorem 1). Then the symbol-pair distance d sp (C 7 ) of C 7 is given by Proof. First of all, since u 2 (x − γ 0 ) W ∈ C 7 , it follows that Now, consider an arbitrary polynomial c(x) ∈ C 7 . We consider two cases. * Case 1: c(x) ∈ u . In this case, by (1). We have In this case, by (1). We have Now by applying Theorem 3, we obtain the desired result.
Finally, we determine the symbol-pair distance of Type 8 γ-constacyclic codes.
be a γ-constacyclic codes of length p s over R of Type 8 (as classified in Theorem 1). Then the symbol-pair distance d sp (C 8 ) of C 8 is given by Consider an arbitrary polynomial c(x) ∈ C 8 \ u 2 (x − γ 0 ) c . Now, we consider two cases as follows: * Case 1: c(x) ∈ u . In this case, by (1), we have wt sp (c(x)) ≥ wt sp (uc(x)) . * Case 2: c(x) / ∈ u . In this case, by (1), we have On the other hand, we have that . Now by applying Theorem 3, we obtain the desired result.

Examples
In this section, we present some examples of symbol-pair distances of constacyclic codes of length p s over F p m + uF p m + u 2 F p m (u 3 = 0).
is 0 or a unit in R γ , and then symbol-pair distance, d sp (C 3 ), is given in Table 2.
is 0 or a unit in R γ , then the symbol-pair distance, d sp (C 4 ), is determined in Table 3.
Then the symbol-pair distance, d sp (C 5 ), is given in Table 4.
are 0 or are units in R γ , then the symbol-pair distance, d sp (C 6 ), is determined in Table 5.
are 0 or are units in R γ . Then the symbol-pair distance, d sp (C 7 ), is given in Table 6.