On Symbol-Triple Distance of a Class of Constacyclic Codes of Length 3ps Over Fpm + uFpm

Let <inline-formula> <tex-math notation="LaTeX">$p\not =3$ </tex-math></inline-formula> be any prime. In this paper, we completely determine symbol-triple distance of all <inline-formula> <tex-math notation="LaTeX">$\gamma $ </tex-math></inline-formula>-constacyclic codes of length <inline-formula> <tex-math notation="LaTeX">$3p^{s}$ </tex-math></inline-formula> over the finite commutative chain ring <inline-formula> <tex-math notation="LaTeX">${\mathcal{ R}}=\mathbb F_{p^{m}}+u\mathbb F_{p^{m}}$ </tex-math></inline-formula>, where <inline-formula> <tex-math notation="LaTeX">$\gamma $ </tex-math></inline-formula> is a unit of <inline-formula> <tex-math notation="LaTeX">$\cal R$ </tex-math></inline-formula> which is not a cube in <inline-formula> <tex-math notation="LaTeX">$\mathbb {F}_{p^{m}}$ </tex-math></inline-formula>. We give the necessary and sufficient condition for a symbol-triple <inline-formula> <tex-math notation="LaTeX">$\gamma $ </tex-math></inline-formula>-constacyclic code to be an MDS symbol-triple code. Using that, we establish all MDS symbol-triple <inline-formula> <tex-math notation="LaTeX">$\gamma $ </tex-math></inline-formula>-constacyclic codes of length <inline-formula> <tex-math notation="LaTeX">$3p^{s}$ </tex-math></inline-formula> over <inline-formula> <tex-math notation="LaTeX">$\cal R$ </tex-math></inline-formula>. Some examples of the symbol-triple distance of <inline-formula> <tex-math notation="LaTeX">$\gamma $ </tex-math></inline-formula>-constacyclic codes of length <inline-formula> <tex-math notation="LaTeX">$3p^{s}$ </tex-math></inline-formula> over <inline-formula> <tex-math notation="LaTeX">$\cal R$ </tex-math></inline-formula> are provided. We also list some new MDS symbol-triple <inline-formula> <tex-math notation="LaTeX">$\gamma $ </tex-math></inline-formula>-constacyclic codes of length <inline-formula> <tex-math notation="LaTeX">$3p^{s}$ </tex-math></inline-formula> over <inline-formula> <tex-math notation="LaTeX">$\cal R$ </tex-math></inline-formula>, where <inline-formula> <tex-math notation="LaTeX">$\gamma $ </tex-math></inline-formula> is not a cube in <inline-formula> <tex-math notation="LaTeX">$\mathbb {F}_{p^{m}}$ </tex-math></inline-formula>.


I. INTRODUCTION
The class of constacyclic codes is an important class of linear codes in coding theory. Many optimal linear codes are directly derived from constacyclic codes. Constacyclic codes have practical applications as they are effective for encoding and decoding with shift registers.
λ-constacyclic codes of length n over F are classified as the ideals ⟨g(x)⟩ of the ambient ring F[x] ⟨x n −λ⟩ where g(x) is a divisor of x n − λ and λ is a unit in the finite field F p m . If (n, p) = 1, the code is called a simple-root code. Otherwise, it is called repeated-root code. Repeat-root codes were studied earlier from the 1960s in some papers (for examples, [1], [2], [24], [25], and [27]). Since the last decade, repeated-roots codes have received much more attention as there have been many more optimal codes obtained from this class of codes. Dinh ([6], [8], [9], [10], [11]) determined the algebraic structures The associate editor coordinating the review of this manuscript and approving it for publication was Yu Zhang. of constacyclic codes in terms of generator polynomials over F p m of length mp s , where m = 1, 2, 3, 4, 6.
In 2011, Cassuto and Blaum ( [3], [4]) introduced a new metric, called symbol-pair metric. Let σ be the code alphabet consisting of q elements. Then each element v ∈ σ is called a symbol. In symbol-triple read channels, a codeword (v 0 , v 1 , . . . , v n−1 ) is read as . . , (v n−1 , v 0 , v 1 )). A q-ary code of length n is a nonempty subset C ⊆ σ n . Assume that v = (v 0 , v 1 , . . . , v n−1 ) is a codeword in σ n . The symbol-triple codeword of v is defined as Hence, each vector has a unique symbol-triple codeword γ (v) ∈ (σ, σ, σ ) n . The symbol-triple distance is an important parameter of symbol-triple codes. Given v = (v 0 , v 1 , . . . , v n−1 ), t = (t 0 , t 1 , . . . , t n−1 ), the symbol-triple distance between v and t is defined as In 2008, the Hamming distance of all cyclic codes of prime power lengths over F p m is given by Dinh [6]. In 2010, [7] computed the Hamming distance of all (α + uβ)-constacyclic codes of length p s over R = F p m +uF p m . After that, the Hamming distance of all constacyclic codes of length 3p s over F p m is provided in [14]. In addition, the Hamming distance of all γ -constacyclic codes of prime power lengths over R is studied in [18]. In 2020, the Hamming distance of λ-constacyclic codes of length 3p s over R is established in R [15], where λ = α + uβ is not a cube. In 2020, the Hamming distances and b-symbol distances of λ-constacyclic codes of length 4p s over R are determined for p m ≡ 1 (mod 4) and the non-square unit λ [16]. In this paper, we completely symboltriple distance of λ-constacyclic codes of length 3p s over R, where λ is not a cube in F p m . In addition, we determine all MDS symbol-triple codes. As an application, some new MDS symbol-triple codes are given. Note that the structure of codes of length 3p s is much more complicated than codes of length 4p s . Repeated-root constacyclic codes of length 3p s over R form a very interesting class of constacyclic codes. When λ is not a cube in F p m , symbol-triple distance of λ-constacyclic codes of length 3p s over R did not study in the past.
Motivated by these, we determine symbol-triple distance of λ-constacyclic codes of length 3p s over R, where λ is not a cube in F p m in this paper. As an application, we identify all the MDS symbol-triple codes among such codes. We also give some new MDS symbol-triple codes.
The rest of this paper is organized as follows. Section II gives some preliminaries. Section III obtains the symbol-triple distance of all γ -constacyclic codes of length 3p s over R (λ is not a cube in F p m ). In Section IV, we give the necessary and sufficient condition for a symbol-triple γconstacyclic code to be an MDS symbol-triple code and we identify all such codes. Some new MDS symbol-triple codes are provided in Section IV. The conclusion of this paper is given in Section V.

II. PRELIMINARIES
For a unit λ of R, the λ-constacyclic (λ-twisted) shift ρ λ on R n is the shift and a code C is said to be λ-constacyclic if ρ λ (C) = C. If λ = {1, −1}, then C is a cyclic and negacyclic code, respectively. Proposition 1: [23] Let C be a linear code. Then C is a λ-constacyclic code of length n over R if and only if C is an ideal of the ring R[x] ⟨x n −λ⟩ . Proposition 2: [17] The dual of a λ-constacyclic code is a λ −1 -constacyclic code.
Let p be a prime and R be a finite chain ring of size p m . Proposition 3: [23] Let C be a linear code C of length n over R. Then |C| = p k , for some integer k ∈ {0, 1, . . . , mn}. In addition, |C| · |C ⊥ | = |R| n , where C ⊥ is the dual code of C.
Assume that α and β are elements in F p m . It is easy to see that α + uβ is an invertible element over R if and only if α ̸ = 0. Therefore, we divide all λ-constacyclic codes of length 3p s over R into the following cases: λ is a cube and p m ≡ 1 (mod 3), λ is a cube and p m ≡ 2 (mod 3), λ = α + uβ is not a cube and 0 ̸ = α, β ∈ F p m , λ is not a cube and 0 ̸ = λ ∈ F p m . We give all λ-constacyclic codes of length 3p s over R in the following theorem.
Theorem 1: [15] Let p ̸ = 3 be any prime. Let C be a λconstacyclic code of length 3p s over R.
1) Assume that λ is a cube in R and p m ≡ 1 (mod 3). Let λ 0 ∈ R such that λ 3 0 = λ and δ, θ ∈ F p m such that δθ = 1 and δ + θ = −1. Then C = C 1 ⊕ C 2 ⊕ C 3 where C 1 is a λ 0 -constacyclic code of length p s over R, C 2 is a δλ 0 -constacyclic code of length p s over R and C 3 is a θ λ 0 -constacyclic code of length p s over R. In particular, |C| =
• Type 2: (principal ideals with nonmonic polynomial generators) In addition, the number of codewords of C, denoted by n C , is given as follows: • If C = ⟨0⟩ and C = ⟨1⟩, then n C = 1 and n C = p 6mp s , respectively.
Then the b-symbol distance between two codeword v and t in σ n is denoted by d b (v, t) and defined as Recently, Yaakobi et al. [26] generalized the coding framework for symbol-pair read channels to that for b-symbol read channels, where the read operation is performed as a consecutive sequence of b > 2 symbols. They also generalized some of the known results for symbol-pair read channels to those for b-symbol read channels. In [21], Dinh et For symboltriple distance, we have the following theorem.

III. SYMBOL-TRIPLE DISTANCE
In [13], the authors obtained the symbol-pair distances of all constacyclic codes of prime power lengths over F p m . Later, [18] and [20] gave the symbol-pair distances of all constacyclic codes of length p s over F p m + uF p m . In this section, when γ is not a cube in F p m , we determine the symbol-triple distance of all γ -constacyclic code of length 3p s over R, where the structure of γ -constacyclic codes of length 3p s over R is given in part 4 of Theorem 1. Denote Then d st (C) can be determined as follows.
Theorem 3: is given by Theorem 2 as follows: The symbol-triple distance of γ -constacyclic codes of Type 3 of length 3p s over R is provided in the following theorem.
Theorem 4: Hence, Proof: Since T is the smallest integer satisfying u(α(x)) T ∈ ⟨(α(x)) j + u(α(x)) r v(x)⟩, we see that Let c(x) ∈ C be an arbitrary polynomial. Then there are two polynomials f 0 (x) and f u (x) over F p m satisfying Now, we consider two cases, namely, v(x) = 0 and v(x) ̸ = 0.
Case 2: Assume that v(x) ̸ = 0. Then we see that . □ We compute the symbol-triple distance of γ -constacyclic codes of Type 4 in the following theorem.
Theorem 5: Let C = ⟨(α(x)) j +u(α(x)) r v(x), u(α(x)) ω ⟩ be a γ -constacyclic code of Type 4 of length 3p s over R, where α(x) = x 3 − γ 0 , v(x) is same as given in Type 3, deg(v) ≤ ω − r − 1, ω < T , and T is the smallest integer satisfying u(α(x)) T ∈ ⟨(α(x)) j + u(α(x)) r v(x)⟩, i.e., T = j, if v(x) = 0 and otherwise T = min{j, p s −j+t}. Then we have d st (C) = d st (⟨(α(x)) ω ⟩ F ), and is given by We need to prove that d st (⟨(α(x)) ω ⟩ F ) ≤ d st (C). In order to do this, let c(x) ∈ C be an arbitrary polynomial and we will prove that wt st (c(x)) ≥ d st (⟨(α(x)) ω ⟩ F ). We see that there exist polynomials f 0 (x), f u (x), g 0 (x) and g u (x) over F p m satisfying where is not a cube in R, then there is an α 1 ∈ F p m satisfying α = α p s 1 . As in part 3 of Theorem 1, (α + uβ)-constacyclic codes of length 3p s over R are the ideals is not a cube in R, we determine the symbol-triple distance of all (α + uβ)-constacyclic codes of length 3p s over R in the following remark.
Proof: We consider three cases. Case 1: If j = 0 and j = 2p s , then C = ⟨1⟩ and C = ⟨0⟩. It is easy to verify that d st (C) = 3 and d st (C) = 0, respectively.
. By Theorem 2, we can determine the symbol-triple distance of ⟨( We present some examples of symbol-triple distance γ -constacyclic codes of length 3p s over F p m + uF p m , where γ ∈ F * p and γ is not a cube. In Table 1, we compute the symbol-triple distances for p = 7, m = 1, s = 1 and 2 and in Table 2, symbol-triple distances have been computed by taking p = 13, m = 1, s = 1 and 2.

IV. MDS SYMBOL-TRIPLE CODES
In 2018, Ding et al. [5] discussed the Singleton bound with respect to d b (C). Following them, the Singleton bound with respect to the b-symbol distance is given as |C| ≤ q n−d b (C)+b . For symbol-triple distance, we need to have the following result. Next, we give all symbol-triple MDS codes of length 3p s over R when λ is a unit of the form λ = γ ∈ F * p m and λ is not a cube. First, we consider the symbol-triple γ -constacyclic code C of length 3p s over R, where C is a symbol-triple γconstacyclic code of Type 1 of length 3p s over R, i.e., C = ⟨0⟩ and C = ⟨1⟩.
Theorem 7: Let C be a symbol-triple γ -constacyclic code of Type 1 of length 3p s over R. Then C = ⟨1⟩ is an MDS symbol-triple code.
Case 2: If C = ⟨1⟩, then d st (C) = 3. Hence, C is an MDS symbol-triple code when |C| = p 2m(3p s −d st (C)+3) , i.e., p 6mp s = p 2m(3p s ) , which is true for all p and s. Therefore, the code C = ⟨1⟩ is an MDS symbol-triple code. □ We determine the MDS condition for symbol-triple γconstacyclic codes of Type 2 of length 3p s over R.
Theorem 8: Let C = ⟨u(x 3 − γ 0 ) j ⟩ be a symbol-triple γconstacyclic code of Type 2 of length 3p s over R, where 0 ≤ j ≤ p s − 1. Then C is not an MDS symbol-triple γconstacyclic code.
Thus, C is not an MDS symbol-triple γ -constacyclic code in this case. □ In the following, we consider the MDS condition for symbol-triple γ -constacyclic codes of Type 3 of length 3p s over R.
Theorem 9: ⟨x 3p s −γ ⟩ or 0. Then C is an MDS symbol-triple γ -constacyclic code of Type 3 of length 3p s over R if one of the following conditions holds true: Now we see that It implies that C is an MDS symbol-triple γ -constacyclic code of Type 3 of length 3p s over R if and only if s = 1 (in such case, j = δ, d st (C) = 3(δ + 1)), or δ = 1, ξ = 0 (in such case, j = 1, d st (C) = 6), or Therefore, C is not an MDS symbol-triple γ -constacyclic code of Type 3 of length 3p s over R. □ Finally, we determine the MDS condition for symbol-triple γ -constacyclic codes of Type 4 of length 3p s over R.
Hence, 3j > d st (C) − 3. Thus, C is not an MDS symbol-triple code. □ Example 2: We provide some new MDS symbol-triple γconstacyclic codes of length 3p s over F p m + uF p m , where γ ∈ F * p and γ is not a cube as follows.

V. CONCLUSION
The metrics of constacyclic codes have very significant role in error-correcting coding theory. In [15], we completely determined the Hamming distance of γ -constacyclic codes of length 3p s over R, where γ is not a cube in F p m . In this paper, the symbol-triple distances of all γ -constacyclic codes of length 3p s over R, where γ is not a cube in F p m are determined (Theorems 3-5). The symbol-triple distance of (α + uβ)-constacyclic codes of length 3p s over R is given in Remark 1, where (α+uβ) is not a cube in R. Example 1 gives us some examples of symbol-triple distance γ -constacyclic codes of length 3p s over R, where γ is not a cube in F p m . We provide the necessary and sufficient conditions for MDS symbol-triple codes of length 3p s over R (Theorems 7-10 and Remark 2). Some new MDS symbol-triple γ -constacyclic codes of lengths 21, 39, 57 over F 7 + uF 7 , F 13 + uF 13 and F 19 + uF 19 are shown in Example 2. For future work, it will be very interesting to study symbol-triple distance of λ-constacyclic codes of length 3p s over R, where λ is a cube in R. In a near future, we will discuss the b-symbol metrics for all constacyclic codes of length 3p s over R and as an application, we will identify all MDS constacyclic codes of length 3p s with respect to bsymbol distances.