b-Symbol Distance of Constacylic Codes of Length ps Over Fpm + uFpm

In this research paper, the repeated-root constacyclic codes over the chain ring <inline-formula> <tex-math notation="LaTeX">$\mathcal F_{p^{m}}+ u \mathcal F_{p^{m}}$ </tex-math></inline-formula> are considered, where <inline-formula> <tex-math notation="LaTeX">$p$ </tex-math></inline-formula> is a prime and <inline-formula> <tex-math notation="LaTeX">$m > 0$ </tex-math></inline-formula> is any integer. The <inline-formula> <tex-math notation="LaTeX">$b$ </tex-math></inline-formula>-symbol distance for prime power length, i.e. <inline-formula> <tex-math notation="LaTeX">$p^{\mathfrak {s}}$ </tex-math></inline-formula> is also studied for any integer <inline-formula> <tex-math notation="LaTeX">${\mathfrak {s}} > 0$ </tex-math></inline-formula>. The Hamming and symbol-pair distances of all <inline-formula> <tex-math notation="LaTeX">$\delta $ </tex-math></inline-formula>-constacyclic codes have been thoroughly studied in <xref ref-type="bibr" rid="ref18">[18]</xref>, where <inline-formula> <tex-math notation="LaTeX">$\delta $ </tex-math></inline-formula> is an unit in the ring <inline-formula> <tex-math notation="LaTeX">$\mathcal F_{p^{m}}+ u \mathcal F_{p^{m}}$ </tex-math></inline-formula> which is of the form <inline-formula> <tex-math notation="LaTeX">$\zeta $ </tex-math></inline-formula> and <inline-formula> <tex-math notation="LaTeX">$\phi + u \varphi $ </tex-math></inline-formula>, where <inline-formula> <tex-math notation="LaTeX">$0 \neq \phi, \varphi, \zeta \in \mathcal F_{p^{m}}$ </tex-math></inline-formula>. In this paper, the <inline-formula> <tex-math notation="LaTeX">$b$ </tex-math></inline-formula>-symbol distance of all such <inline-formula> <tex-math notation="LaTeX">$\delta $ </tex-math></inline-formula>-constacyclic codes of prime power length is computed for <inline-formula> <tex-math notation="LaTeX">$1 \leq b \leq \lfloor \frac {p}{2}\rfloor $ </tex-math></inline-formula>. Furthermore, as an application, all MDS <inline-formula> <tex-math notation="LaTeX">$b$ </tex-math></inline-formula>-symbol constacyclic codes of length <inline-formula> <tex-math notation="LaTeX">$p^{\mathfrak {s}}$ </tex-math></inline-formula> over <inline-formula> <tex-math notation="LaTeX">$\mathcal F_{p^{m}}+ u \mathcal F_{p^{m}}$ </tex-math></inline-formula> are established.

The theory of error correcting codes attracted the attention of many researchers around the globe since its eruption. At the beginning, the message in the noisy channel was split up into information units called symbols. Researchers used to perform the operations of writing and reading on these individual symbols, which created a lot of disruptions. These conditions were sorted out with the introduction of symbols that can be read and written in possible overlapping groups. Cassuto and Blaum [1] were the first to propose this method in which the outputs of the channel is overlapping pair of symbols. The model proposed by Cassuto and Blaum [1] and later by Cassuto and Litsyn [2] deals with the possibly corrupted outputs generated by a string of read operations having overlapping pairs of adjoining symbols, called pairread symbols. Later, Kai et al. [20] developed the theory given by Cassuto and Litsyn [3,Th. 10] for simple-root constacyclic codes. Many researchers scrutinized symbol-pair distances over constacyclic codes since then in [11], [14], [16]- [18], [21] over many years.
As a result of rich algebraic structure and practical implementations, constacyclic codes play a remarkable role in The associate editor coordinating the review of this manuscript and approving it for publication was Xueqin Jiang . coding theory. Repeated-root constacyclic codes were inducted by Castagnoli [4] and Van Lint [22], where they established that the repeated-root constacyclic codes have a sequential structure. The researchers like Cao further established some significant results over repeated-root constacyclic codes in [5]- [10]. But the existence of optimal repeated-root constacyclic codes influenced the researchers to put these codes under further scrutiny.
The results established for symbol-pair read channels were further generalized to b-symbol read channels by Yaakobi et al. [23], in which the read operation is performed on overlapping b-pairs of adjacent symbols, where b ≥ 3.
In [12], Ding  where is an alphabet consisting symbols of size q and x = (x 0 , x 1 , . . . , x n−1 ) is a vector in n . The b-symbol distance between vectors y and x in n is denoted by d b (y, x) and defined as A Singleton Bound for b-symbol codes over F q was established in [12] as follows: Let q be a prime power and b ≤ d b ≤ n, for any b-symbol code C of length n with size M and minimum b-distance d b over F q , M ≤ q n−d b +b . The b-symbol code C is called an optimal code or maximum distance separable (MDS) b-symbol code, if the equality holds, with respect to Singleton bound. Some families of linear MDS b-symbol codes over finite field for some special b are also constructed.
The Hamming and symbol-pair distances over F p m + uF p m and b-distances of over F p m of repeated root constacyclic codes of prime power lengths have been computed by Dinh et al. [18], [19], recently. These works motivate us to study b-symbol distances of repeated root constacyclic codes of p s length over F p m + uF p m , u 2 = 0. We determine b-distance of all the constacyclic codes of length p s over F p m + uF p m for 1 ≤ b ≤ p 2 . As an application, we establish all MDS b-symbol constacyclic codes of length p s over the ring F p m + uF p m .
The organization of this paper is as follows. Some preliminary results are discussed in Section 2. In Section 3, the b-distance of constacyclic codes of length p s is established over the ring F p m +uF p m . All the MDS b-symbol constacyclic codes of length p s are identified in Section 4 and in Section 5, we conclude the paper.
Consider the polynomial c(x) = c 0 + c 1 x + c 2 x 2 + . . . + c n−1 x n−1 in the ring R[x]/ x n − δ . The polynomial c(x) can be used to express the codeword c = (c 0 , c 1 , . . . , c n−1 ) of the code C. And xc(x) corresponds to δ-constacyclic shift of c(x). We have a well-known result about δ-constacyclic codes.
So, for any unit δ of F p m , the linear δ-constacyclic code of length p s over F p m are precisely the ideals in F p m [x]/ x p s − δ . From Division Algorithm, there exists non-negative integers k q , k r such that s = k q m + k r , and 0 ≤ k r ≤ m − 1. Let where 0 ≤ ι ≤ p s , which forms the strictly inclusive chain − δ form the desired strictly inclusive chain. Clearly, for ι = 0, 1, . . . , p s , the cardinality of each code . Theorem 2.3: The dual of the δ-constacyclic codes is the δ −1 -constacyclic codes over F p m and is given by Hamming weight of a codeword a = (a 0 , a 1 , . . . , a n−1 ) ∈ C is the non-zero entries in a and is denoted by wt H (a). The no. of non-zero entries in the codeword a − b = (a 0 − b 0 , a 1 − b 1 , . . . , a n−1 − b n−1 ) is the Hamming distance between two codewords a = (a 0 , a 1 , . . . , a n−1 ) and b = (b 0 , b 1 , . . . , b n−1 ) ∈ C and is denoted by d H (a, b). The Hamming distance of all δ-constacyclic code depends on the characteristic of F p m and length of the code.
Theorem 2.4: Hamming distance of the constacyclic code = 0). The elements of R can be written as a + ub, where a, b ∈ F p m . The construction of all constacyclic codes of prime power length over R is provided by Dinh in [13] as follows. VOLUME 8, 2020 Theorem 2.5: Let δ be a unit of the ring R, i.e. δ is of the form φ + uϕ or ζ , where 0 = φ, ϕ, ζ ∈ F p m .
When the unit δ is of the form φ + uϕ, d H (C) of each (φ + uϕ)-constacyclic codes of p s length over R were provided in [13]. If the unit δ is of the form ζ , then d H (C) of each ζ -constacyclic codes is given in [18]. We note that F p m is a subring of R, for a code C over R, we denote d H (C F ) as the Hamming distance of C| F p m . For each codeword c = (c 0 , c 1 , . . . , c n−1 ) over R, its polynomial representative c(x) is given as As we know that Type 1 consists of trivial ideals 0 , 1 , their Hamming distance are 0 and 1 respectively. The Hamming distance for the ideals of the form u(x − ζ 0 ) ι , 0 ≤ ι ≤ p s − 1 is given by the following theorem.
, Hamming distance is given by the following theorem.
Theorem 2.8: Let C be a ζ -constacyclic codes of length p s over R of Type 3,i Hamming distance is given by the following theorem.

III. b-SYMBOL DISTANCE
Recently, the b-symbol distance distribution of all constacyclic code of length p s over F p m has been discussed by Dinh et al. in [19].
The b-symbol distances of all δ-constacyclic codes of length p s over the ring R is computed in this section. Let δ = φ + uϕ be the unit, where φ, ϕ ∈ F * p m , i.e., (φ + uϕ)-constacyclic codes of the length p s over R.
. We have two cases: x p s −φ are φ-constacyclic codes of length p s over F p m , whose b-symbol distances are computed in Theorem 3.1.
Thus, following the methodology of [18] we obtain the b-symbol distances of each (φ + uϕ)-constacyclic codes of p s length over R are given below: Now, we consider the case where the unit is of the form )}. Thus, we provide the b-symbol distances of all ζ -constacyclic codes of length p s over R, for ζ ∈ F * p m by following the methodology given in [18]. Theorem 3.3: Let δ = ζ ∈ F * p m be a unit in R. The ζ -constacyclic codes of length p s over R have their b-symbol distances completely determined as follows.
• Type 1 trivial ideals: • Type 2 principal ideals generated by non-monic polynomial: • Type 3 principal ideals generated by monic polynomial: • Type 4 non-principal ideals: ; that is T can be determined as: Then It can be noted that in Theorem 3.2 and 3.3, when b = 1 and b = 2, the b-distance gives the Hamming distance and the symbol-pair distance of all the δ-constacyclic codes of length p s over R = F p m +uF p m , respectively. It can be observed that the resultant Hamming and symbol-pair distances are similar to the distances deduced in [18]. It can also be inferred that the Hamming distances can be obtained for p ≥ 2, whereas the symbol-pair distances can be acquired for p ≥ 5. Now, let us consider an example of b-symbol constacyclic codes of length p s over F p m + uF p m . For (φ + uϕ)-constacyclic code of length 49 over R the generators are of the form C ι = (φ 0 x − 1) ι , for ι ∈ {0, 1, . . . , 98}. Then the 3-symbol distance is determined in Table 1. Now, ζ -constacyclic codes of length 49 over R has four types of generator. The distances corresponding to different generators are given as follows:  • Type 2 (principal ideals generated by nonmonic polynomial ): Table 2.

IV. MDS b-SYMBOL CONSTACYCLIC CODES OF LENGTH p s OVER F p m + uF p m
All the MDS b-symbol constacyclic codes are identified in this section by utilizing the b-symbol distance obtained in Section III of all b-symbol constacyclic codes of prime power length over F p m + uF p m .
Hence, no MDS code can be obtained here.
Thus, in this case there is no MDS code.
Therefore, in this case there is no MDS b-symbol constacyclic code.
Thus, the only MDS b-symbol (φ +uϕ)-constacylcic codes of length p s over R is the trivial code 1 . Now, the case where the unit δ = ζ ∈ F * p m is considered. From [13], it is acquired that for a ζ -constacyclic code, four types of ideals can be obtained and the dimension of the code C ι varies with each ideal. Here, the b-symbol MDS codes for each type of ideal is determined, whose b-symbol distance has been discussed in Theorem 3.3.
Thus, there is only one b-symbol codes for trivial ideals, i.e., 1 .

2) TYPE 2 (PRINCIPAL IDEALS GENERATED BY NONMONIC POLYNOMIAL)
Here, we have C = u(x − ζ 0 ) ι , where 0 ≤ ι ≤ p s − 1 and |C| = p m(p s −ι) . Thus by Singleton bound, C is a b-symbol MDS code if and only if p Hence, we have the following theorem.
x p s −ζ be a ζ −constacyclic code of length p s over R, for ι ∈ 0, 1, . . . , p s − 1. Then C is not a MDS b-symbol constacyclic code.
Proof: We get MDS code for ι = 2d b (C) − p s − 2b. Now, we consider the cases according to the range of ι.
Case 1: ι = 0, then d b (C) = b, and the Singleton bound is satisfied when p s = 0, which is a contradiction. Thus, no MDS b-symbol constacyclic code can be obtained in this case.

in this case no MDS b-symbol constacyclic code can be obtained.
This completes the proof.

3) TYPE 3 (PRINCIPAL IDEALS GENERATED BY MONIC POLYNOMIAL)
Here, x p s −ζ . Thus, the following two cases can be obtained: Hence, the MDS codes for these ideals are similar to the MDS codes obtained for δ-constacyclic b-symbol codes over F p m . Hence, we have the following theorem: x p s −ζ be a ζ −constacyclic code of length p s over R, for ι ∈ {1, . . . , p s − 1}. Then C is a MDS b-symbol code if and only if one of the following conditions holds: [13] then, Therefore, when 1 ≤ ι ≤ p s−1 + τ 2 , MDS b-symbol constacyclic codes can be obtained when ι = d b (C) − b. Hence, the MDS codes for these ideals are similar to the MDS codes obtained in the above case. But, we have 2i−2p s−1 ≤ τ and 0 ≤ τ < ι, i.e., p s−1 ≤ ι < 2p s−1 . Thus, when s = 1, ι = 1 and when s ≥ 2, 2 ≤ ι ≤ 2p s−1 . Again, p s −b < 2p s−1 if and only if p = 2. Hence, we conclude the theorem as follows.
Theorem 4.4: x p s −ζ be a ζ −constacyclic code of length p s over R, for 1 ≤ ι ≤ p s−1 + τ 2 . Then C is a MDS b-symbol code if and only if one of the following conditions holds: In the following theorem we discuss the case when

. Then, C is a MDS b-symbol code if and only if one of the following conditions:
For p = 2, the condition for MDS code is modified to 2ι > 2d b (C) − 2b and τ ≥ 0. Now, by considering d b (C) for different ranges of ι from Theorem 3.3 we have the following cases: +p s−1 (p − 2) + 2b + 2θ. Now, we consider two sub-cases: Thus, MDS b-symbol constacyclic code can be obtained when θ > 2−b, which is a contradiction, since we have the equality θ + b = 2.
Thus, MDS b-symbol constacyclic code can be obtained with equality = s−2, ψ = p−2 and θ +b = p. But θ(ψ +1) ≤ b gives p 2 ≤ b + 1, which is a contradiction. Thus, MDS b-symbol constacyclic code can not be obtained in this case.
which is a contradiction. Therefore, no MDS b-symbol constacyclic code can be obtained in this case.

Thus, the equality occurs at
Hence, all the conditions for MDS code are satisfied.

MDS b-symbol constacyclic code exists in this case.
This completes the proof.

4) TYPE 4 (NONPRINCIPAL IDEALS)
Here, In this case, |C| = p m(2p s −ι−κ) . Thus by Singleton bound, C is a b-symbol MDS code if and only if 2p Thus, the condition for C to be a b-symbol MDS constacyclic code becomes κ = 2d b (C) − 2b − p s + η. Hence, we can conclude the following theorem: x p s −ζ be a ζ −constacyclic code of length p s over R, for ι ∈ 1, . . . , p s − 1, 0 ≤ τ < ι, and either h(x) is either 0 or a unit in Then C is not a MDS b-symbol constacyclic code. Proof: We get MDS code for κ = 2d b (C) − p s − 2b + η, where 1 ≤ η ≤ p s − 1 and κ < ι. Now, we consider the cases according to the range of κ.
Clearly, ι < κ, which is a contradiction. Hence, no MDS b-symbol constacyclic code can obtained in this case.
This completes the proof. Consequently, we have the list of all MDS b-symbol constacyclic codes of length p s over R = F p m + uF p m .
Then all MDS b-symbol δ-constacyclic codes of length p s over R are determined as follows: • For ζ -constacyclic codes, there are four types of ideals: • Type 1 (trivial ideals): 1 is the only b-symbol constacyclic code with d b (C) = b.

No MDS b-symbol constacyclic codes can be obtained in this case.
• Type 3 (principal ideals generated by monic poly- x p s −ζ . When h(x) = 0, then C is a MDS b-symbol code if and only if one of the following conditions holds: Then C is a MDS b-symbol code if and only if one of the following conditions holds: is a unit and p s−1 + τ 2 < ι ≤ p s − 1. Then, C is a MDS b-symbol code if and only if one of the following conditions holds: • Type 4 (nonprincipal ideals): Here, we consider b = 3, i.e., 3-symbol δ-constacyclic codes of length 49 over F 7 +uF 7 , where p = 7, m = 1, s = 2 and δ is a unit in F 7 +uF 7 of the form φ+uϕ and ζ , with φ, ϕ, ζ ∈ F * 7 . In example 3.4, we determined all the 3-symbol distances for all the ranges of ι and κ. From the values of ι, κ and d 3 (C), we have the following list of MDS 3-symbol δ-constacyclic codes.
Here, the condition for MDS code is given by ι = 2d 3 (C) − 6 and the only MDS code obtained is 1 with d 3 ( 1 ) = 3.
• Type 2 (principal ideals generated by the non monic polynomial): C = u(x − ζ 0 ) ι where 0 ≤ ι ≤ 48. MDS 3-symbol codes for these codes are obtained by the condition ι = 2d 3 (C) − 55, which is not satisfied by any value of ι and d 3 (C). Thus, no MDS code is obtained in this case.

V. CONCLUSION
In this research article, all b-symbol distances of repeatedroot constacyclic codes having length p s have been determined over F p m + uF p m , influenced by the notion of Dinh et al. [18], [19]. The b-symbol distances over F p m + uF p m are quite similar to those over F p m , with variable range of ι for each type of ideal. All MDS b-symbol codes are determined as an application among repeated-root constacyclic codes of p s length over the ring F p m + uF p m .
We can further generalize these results for computing b-symbol distances of constacyclic codes of length 2p s over F p m and consequently on R. It will be interesting to obtain some more optimal codes.