Abstract
Let p be a prime with \(5 |(p-1)\). Let S be a set of all repeated-root cyclic codes \({\mathcal {C}}=\langle g(x)\rangle \), \((x^5-1)|g(x)\), of length 5p over a field field \({\mathbb {F}}_p\), whose Hamming distances are at most 7. In this paper, we present a method to find all maximum distance separable (MDS) symbol-pair codes in S. By this method we can easily obtain the results in Ma and Luo (Des Codes Cryptogr 90:121–137, 2022) and new MDS symbol-pair codes, so we remain two possible MDS symbol-pair codes for readers.
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1 Introduction
Symbol-pair codes introduced by Cassuto and Blaum [1] are designed to protect against pair errors in symbol-pair read channels. Cassuto and Litsyn [3] constructed cyclic symbol-pair codes using algebraic methods and showed that there exist symbol-pair codes whose rates are strictly higher, compared to codes for the Hamming metric with the same relative distance. Yaakobi et al. [16] studied b-symbol read channels and generalized some of the known results for symbol-pair codes to those for b-symbol read channels. Dinh et al. [9,10,11] investigated the symbol-pair weight distributions of repeated-root constacyclic codes etc.
The minimum symbol-pair distance plays an important role in determining the error correcting capability of a symbol-pair code. In general, a code over \({\mathbb {F}}_q\) of length n with size M and minimum pair-distance \(d_p\) is called an \((n, M, d_p)\) symbol-pair code. An \((n, M, d_p)\) symbol-pair code can correct up to \(\lfloor (d_p-1)/2\rfloor \) pair errors (see [1, Proposition 3]). Chee et al. [4] gave the Singleton-type bound for symbol-pair codes relates the parameters n, M and \(d_p\).
Lemma 1.1
[4] (Singleton Bound) Let q be a prime power and \(2 \le d_p \le n\). If \({\mathcal {C}}\) is an \((n, M, d_p)\) symbol-pair code over \({\mathbb {F}}_q\), then \( M \le q^{n-d_p+2}\). If \(M = q^{n-d_p+2}\), then it is called an maximum distance separable (MDS) symbol-pair code.
A q-ary MDS symbol-pair code with parameters \((n, M, d_p)\) is simply called an MDS \((n, d_p)\) symbol-pair code.
There are several works that have contributed to the constructions of MDS symbol-pair codes. Chee et al. [4, 5] obtained many classes of MDS symbol-pair codes from classical MDS codes and interleaving method of Cassuto and Blaum [1]. Moreover, they obtained nontrivial MDS symbol-pair codes with length \((q^2+2q)/2\) by employing classical MDS codes and Eulerian graphs of certain girth. Kai et al. [12] constructed MDS symbol-pair codes with \(d_p=5\) based on constacyclic codes. Later Kai et al. [13] derived three families of MDS symbol-pair codes by using repeated-root constacyclic codes. Ding et al. [7] obtained MDS symbol-pair codes with \(d_p=6\), whose lengths from 6 to \(q^2 + 1\), moreover, they found some MDS symbol-pair codes with \(d_p\ge 7\) utilizing elliptic curves. Then they investigated MDS b-symbol codes [8]. Li et al. [14] gave a number of MDS symbol-pair codes with \(d_p=7\) by analyzing some linear fractional transformations. Chen et al. [6] obtained MDS symbol-pair codes with \(d_p=8\) of length 3p from repeated-root cyclic codes. Recently, Ma and Luo [15] constructed two classes of MDS symbol-pair codes with \(d_p=10\) and \(d_p=12\) from repeated-root cyclic codes of length 3p over \({\mathbb {F}}_p\). However, it becomes difficult to find MDS symbol-pair codes possessing comparatively large length and minimum pair-distance.
In this paper, let p be a prime with \(5 |(p-1)\). Let S be a set of all repeated-root cyclic codes \({\mathcal {C}}=\langle g(x)\rangle \), \((x^5-1)|g(x)\), we present a method to find MDS symbol-pair codes of length 5p over \({\mathbb {F}}_p\). Moreover, by the method we can easily obtain the results in [15]. This paper is organized as follows. In Sect. 2, basic notations and results about cyclic codes and symbol-pair codes are provided. In Sect. 3, an unique class of MDS symbol-pair codes with \(d_p=12\) among all repeated-root cyclic codes whose Hamming distance is equal to 6 are investigated. In Sect. 4, we conclude this paper with remarks.
2 Preliminaries
In this section, we review some basic notations, results on cyclic codes, and symbol-pair codes over a finite field, which will be used to prove our main results in the sequel.
2.1 Cyclic code
Let \({\mathbb {F}}_q\) be a finite field with q elements, where \(q=p^s\), p is a prime and s is a positive integer. Let \({\mathcal {C}}\) be an [n, l] linear code over \({\mathbb {F}}_q\), i.e., it is an l-dimensional subspace of \({\mathbb {F}}_{q}^n\). If for each codeword \((c_0,c_1,\ldots ,c_{n-1})\in {\mathcal {C}}\), \(( c_{n-1},c_0,\ldots ,c_{n-2})\) is also in \(\mathcal {C}\), then we call \(\mathcal {C}\) a cyclic code. We identify a codeword \(\textbf{c} =(c_0, c_1, \ldots , c_{n-1})\) in \({\mathcal {C}} \) with the polynomial \(c(x)=c_0+c_1x+c_2x^2+\cdots +c_{n-1}x^{n-1}\) in \({\mathbb {F}}_q[x]/\langle x^n-1\rangle \). A code \({\mathcal {C}} \) of length n over \({\mathbb {F}}_q\) corresponds to a subset of \({\mathbb {F}}_q[x]/\langle x^n-1\rangle \). Then \({\mathcal {C}}\) is a cyclic code if and only if the corresponding subset is an ideal of \({\mathbb {F}}_q[x]/\langle x^n-1\rangle \). Hence there exists a monic divisor g(x) of \(x^n-1\in {\mathbb {F}}_q[x]\) such that
The g(x) is called the generator polynomial of \({\mathcal {C}}\).
A cyclic code is called simple-root cyclic code if \(\gcd (n,p)=1\) and a repeated-root cyclic code if p|n. Castagnoli et al. in [2] studied the Hamming distance of repeated-root cyclic codes by using polynomial algebra, they showed that the Hamming distance of a repeated-root cyclic code \({\mathcal {C}}\) can be expressed in terms of \(d_H(\overline{{\mathcal {C}}}_t)\), where \(\overline{{\mathcal {C}}}_t\) are simple-root cyclic codes fully determined by \({\mathcal {C}}\).
Let \({\mathcal {C}}=\langle g(x)\rangle \) be a repeated-root cyclic code of length \(\ell p^s\) over \({\mathbb {F}}_q\), where \(\ell >1\) is a positive integer such that \(\gcd (\ell ,p)=1\) and s is a positive integer. Suppose that \(g(x)=\Pi _{i=1}^s m_i(x)^{e_i}\) is the factorization of g(x) over \({\mathbb {F}}_q\), where \(m_i(x), i=1,\ldots ,s\) are distinct monic irreducible polynomials of multiplicity \(e_i\). Fixing an integer t, \(0\le t\le p^s-1\), we define \(\overline{{\mathcal {C}}}_t=\langle {\overline{g}}_t(x)\rangle \) a simple-root cyclic code of length \(\ell \) over \({\mathbb {F}}_q\), where \({\overline{g}}_t(x)\) is the product of those irreducible factors \(m_i(x)\) with \(e_i>t\). If this product is equal to \(x^{\ell }-1\), i.e., \(\overline{{\mathcal {C}}}_t\) contains only the zero codeword, then \(d_H(\overline{{\mathcal {C}}}_t)=\infty \). If all \(e_i\) satisfy \(e_i\le t\), then \({\overline{g}}_{t}(x)=1\) and \(d_H(\overline{{\mathcal {C}}}_t)=1\).
The following lemma will be used to determine the Hamming distance of repeated-root cyclic codes \({\mathcal {C}}\), which obtained from [2].
Lemma 2.1
[2] Let \({\mathcal {C}}=\langle g(x)\rangle \) be a repeated-root cyclic code of length \(\ell p^s\) over \({\mathbb {F}}_q\), where p is a prime with \(\gcd (\ell ,p)=1\) and s is a positive integer. Then
where for each \(t\in T=\{t:0\le t\le p^s-1\}\), \(t=t_{0}+t_{1}p+\cdots +t_{s-1}p^{s-1}\) is the p-adic representation and \(P_t=\prod _{m=0}^{s-1}(t_m+1)=w_{H}((x-1)^t)\).
2.2 Symbol-pair codes
For \(x=(x_0,x_1,\ldots ,x_{n-1})\in {\mathbb {F}}_q^n \), the symbol-pair read vector of x is
For a code \({\mathcal {C}}\subset {\mathbb {F}}_q^n\), there is the symbol-pair code generated by \({\mathcal {C}}\):
Let \(x=(x_0, x_1, \ldots , x_{n-1})\) and \(y=(y_0, y_1, \ldots , y_{n-1})\in {\mathbb {F}}_q^n\). Recall that the Hamming weight of the vector x is defined as \(w_H (x) =|\{i: x_i \ne 0, 0\le i\le n-1\}|\) and the Hamming distance between x and y is defined as \(d_H (x,y) =|\{i:x_i \ne y_i, 0\le i\le n-1 \}|\). Define the symbol-pair weight of x as
define the symbol-pair distance between x and y as
where the subscripts \( i+1 \) are reduced modulo n.
An \((n, M, d_p)\) symbol-pair code \(\pi _p({\mathcal {C}})\) generated by \({\mathcal {C}}\subset {\mathbb {F}}_q^n\) has size M and minimum symbol-pair distance \(d_p\), where \(d_p = \min \{d_p(x, y):x, y \in {\mathcal {C}}, x\ne y\}\). Similar to the classical case, if \({\mathcal {C}}\) is a linear code, then the minimum symbol-pair distance of \(\pi _p({\mathcal {C}})\) is the smallest symbol-pair weight of nonzero codewords of \(\pi _p({\mathcal {C}})\), that is
It is known in [1] that for any \(0< d_H({\mathcal {C}}) < n\),
Let \(S=\{(x_i, x_{i+1}): 0\le i\le n-1\}\) be the set from the vector x. There are two subsets of S:
and
It is obvious that \(w_H(x)=|S_0|\) and
where \(L= |S_1|\). In fact if \(x=(x_0,x_1,\ldots , x_{n-1})\in {\mathbb {F}}_q^n\) is viewed as a cycle of length n, then L is the number of a sequence of 0’s in the cyclic of x. For example, in \(x=(1,0,0,1,0,0,0,1,0,1)\) and \(y=(0,1,0,0,1,0,1,0,1,0)\in {\mathbb {F}}_{2}^{10}\), we have \(L=3\) and \(L=4\), respectively.
In this paper, we will utilize repeated-root cyclic codes to obtain a class of new MDS symbol-pair codes. A simple notation is given below.
Definition 2.2
The support of a polynomial \(f(y)=\sum _{i=0}^{\ell -1}a_iy^i\) is the set
and denote the number of elements in supp(f) by N.
3 MDS symbol-pair codes
In this section, we always assume that p is a prime number and \(5|(p-1)\). There is an irreducible factorization over \({\mathbb {F}}_p\):
where \(\zeta \) ia a primitive 5-th root of unity in \({\mathbb {F}}_p\).
Let
be a set of nontrivial cyclic codes with length 5p over \({\mathbb {F}}_p\).
First, we shall find MDS symbol-pair codes from all repeated-root cyclic codes of length 5p with \(d_H({\mathcal {C}})\le 7\) defined as (3.1).
Theorem 3.1
If \({\mathcal {C}}=\langle g(x)\rangle \in S, d_H({\mathcal {C}})\le 7\), and \({\mathcal {C}}\) is an MDS symbol-pair code. Then there is a unique possible code as follows: \( d_H({\mathcal {C}})=6\) and
Proof
Suppose that \({\mathcal {C}}=\langle g(x)\rangle \) is a \([5p,l,d_H({\mathcal {C}})]\) cyclic code with MDS symbol-pair. Then \(d_p({\mathcal {C}})=5p-l+2\) with \(l=5p-\deg (g(x))\), so
In (3.1), \((x^5-1)| {g}(x)\) and \(\deg (g(x))\ge 5\). Recall that \(d_p( {{\mathcal {C}}} )\le 2d_H( {{\mathcal {C}}} )\). Then \(d_H({\mathcal {C}})\ge 4 \).
By Lemma 2.1, \(d_H({\mathcal {C}})=\min \{P_t\cdot d_H(\overline{{\mathcal {C}}}_t): t=1,2,\ldots ,p-1\}\), where \(\overline{ {\mathcal {C}} }_t=\langle {g}_t(x)\rangle \), it is clear that \( {g}_0 (x)= x^5-1 \) and \(P_0\cdot d_H(\overline{{\mathcal {C}}}_0)=\infty \). So we only consider \(1\le t\le p-1\) and \(P_t=t+1\).
(1) Suppose that \(d_H({\mathcal {C}})=4\). Then \(d_p( {{\mathcal {C}}} )\le 8\).
If \(t=1\), then \(d_H(\overline{{\mathcal {C}}}_1)\ge 2\) and \( {g}_1 (x) \) has at least one factor: \(x-1\), this means \( j_0\ge 2\).
If \(t=2\), then \(d_H(\overline{{\mathcal {C}}}_2)\ge 2\) and \( {g}_2 (x) \) has at least one factor: \(x-1\), this means \( j_0\ge 3\).
Thus \(j_0\ge 3 \) and \(j_1\ge j_2\ge j_3\ge j_4 \ge 1\) and \(\deg (g(x))\ge 7\), which is a contradiction.
(2) Suppose that \(d_H({\mathcal {C}})=5\). Then \(d_p({\mathcal {C}})\le 10\).
If \(t=1\), then \(d_H(\overline{{\mathcal {C}}}_1)\ge 3\) and \( {g}_1 (x) \) has at least two factors: \(x-1\) and \(x-\zeta \), this means \( j_0\ge 2\) and \(j_1\ge 2\).
If \(t=2\), then \(d_H(\overline{{\mathcal {C}}}_2)\ge 2\) and \( {g}_2 (x) \) has at least one factor: \(x-1\), this means \( j_0\ge 3\).
If \(t=3\), then \(d_H(\overline{{\mathcal {C}}}_3)\ge 2\) and \( {g}_2 (x) \) has at least one factor: \(x-1\), this means \( j_0\ge 4\).
Thus \(j_0\ge 4\), \(j_1\ge 2\), and \( j_2\ge j_3\ge j_4 \ge 1\), and \(\deg (g(x))\ge 9\), which is a contradiction.
(3) Suppose that \(d_H({\mathcal {C}})=7\). Then \(d_p({\mathcal {C}})\le 14\).
If \(t=1\), then \(d_H(\overline{{\mathcal {C}}}_1)\ge 4\) and \( {g}_1 (x) \) has at least three factors: \(x-1\), \(x-\zeta \), and \(x-\zeta ^2\), this means \( j_0\ge 2\), \(j_1\ge 2\), \(j_2\ge 2\).
If \(t=2\), then \(d_H(\overline{{\mathcal {C}}}_2)\ge 3\) and \( {g}_2 (x) \) has at least two factors: \(x-1\) and \(x-\zeta \), this means \( j_0\ge 3\) and \(j_1\ge 3\).
If \(t=3\), then \(d_H(\overline{{\mathcal {C}}}_3)\ge 2\) and \( {g}_2 (x) \) has at least one factor: \(x-1\), this means \( j_0\ge 4\).
If \(t=4\), then \(d_H(\overline{{\mathcal {C}}}_4)\ge 2\) and \( {g}_2 (x) \) has at least one factor: \(x-1\), this means \( j_0\ge 5\).
If \(t=5\), then \(d_H(\overline{{\mathcal {C}}}_5)\ge 2\) and \( {g}_2 (x) \) has at least one factor: \(x-1\), this means \( j_0\ge 6\).
Thus \(j_0\ge 6\), \(j_1\ge 3\), \(j_2\ge 2\), and \( j_3\ge j_4 \ge 1\), and \(\deg (g(x))\ge 13\), which is a contradiction.
(4) Suppose that \(d_H({\mathcal {C}})=6\). Then \(d_p({\mathcal {C}})\le 12\).
If \(t=1\), then \(d_H(\overline{{\mathcal {C}}}_1)\ge 3\) and \( {g}_1 (x) \) has at least two factors: \(x-1\) and \(x-\zeta \), this means \( j_0\ge 2\) and \(j_1\ge 2\).
If \(t=2\), then \(d_H(\overline{{\mathcal {C}}}_2)\ge 2\) and \( {g}_2 (x) \) at least one factor: \(x-1\), this means \( j_0\ge 3\).
If \(t=3\) and \(t=4\), then either \( {g}_3 (x) \) or \({g}_4 (x) \) has at least one factor: \(x-1\), this means \( j_0\ge 5\).
Thus \(j_0\ge 5 \), \(j_1\ge 2\), and \( j_2\ge j_3\ge j_4\ge 1\). Then
where for \(0\le i\le 4\), \(j_i^\prime \) is a positive integer, and \(\deg (g(x))=10+\sum _{i=0}^4j_i^\prime \).
By (3.3), we have
it can only have
Hence if \({\mathcal {C}}=\langle g(x)\rangle \in S \) and \({\mathcal {C}}\) is an MDS symbol-pair code, then there is a unique possible code as follows: \( d_H({\mathcal {C}})=6\) and
This is completed the proof. \(\square \)
Next, we shall verify that the code in Theorem 3.1 is just MDS symbol-pair with \( d_H({\mathcal {C}})=6\).
Suppose that c(x) is a nonzero code polynomial of \({\mathcal {C}}=\langle g(x)\rangle \in S\). Then g(x)|c(x) and c(x) can be written as the form \(c(x)=\sum _{i=0}^{4} x^iV_i(x^{5}) \), for convenience, we write
where \(V_i(x^{5})\) is a polynomial of \(x^{5}\). Let \(N_i=|supp(V_i(x^5))|, 0\le i\le 4\), where each \(supp(V_i(x^5)\) is in Definition 2.2.
By \(c(1)=c(\zeta )=\cdots =c(\zeta ^{4})=0\), we obtain a system of 5 equations over \({\mathbb {F}}_p\) as follows:
It is easy to check that the coefficient matrix of (3.4) is nonsingular. Then
it is implied that \( (x^{5}-1)|V_{i}(x^{5})\) for each \(0\le i\le 4\). Suppose that \( V_i(x^{5})=\sum _{j=0}^n a_j{(x^5)}^j\), it follows from \( V_{i}(1)=0\) that \( a_0=-(a_1+\ldots +a_n)\).
Theorem 3.2
Let g(x) be defined as (3.2) and \({\mathcal {C}}=\langle g(x)\rangle \). Then \({\mathcal {C}}\) is an MDS symbol-pair codes with \( d_H({\mathcal {C}})=6\).
Now we give some lemmas to prove Theorem 3.2.
Lemma 3.3
If \( w_H(c(x))=6\), then \(w_p(c(x))=12\).
Proof
We divide into three cases to investigate \(w_p(c(x))\) with \( w_H(c(x))=6\).
Case 1: If \(c(x)=(V_i(x^5), V_j(x^5))\) with \((N_i, N_j)=(4,2)\) and \( 0\le i< j\le 4\). Since \(w_H(x^iV_i(x^5))=w_H(V_i(x^5))\), without loss of generality, we consider \(c(x)=(V_0(x^5), V_k(x^5))\) with \(1\le k\le 4\).
Suppose that \( k\in \{2,3\}\). Then \(L=6\) and \(w_p(c(x))=12\).
Suppose that \( k=1 \). Let \(V_0(x^5)=a_0+ a_1x^{5r_1}+a_2x^{5r_2}+a_3x^{5r_3}\) with \(1\le r_1<r_2<r_3<p\) and \(V_1(x^5)=b_1 (x^{5r_4}-1), 1\le r_4<p\). Then
The first, the second and the third formal derivative of c(x) respectively gives
and
Since \((x -1)^5\) and \((x-\zeta )^2\) are divisors of c(x), it follows from \(c^{(1)}(1)=c^{(1)}(\zeta )=c^{(2)}(1)=c^{(3)}(1)=0\), note that \(a_0=-(a_1+a_2+a_3)\), that
where \(B=(B_1,B_2,B_3,B_4)\), and for \(1\le i\le 3\),
and
We make some elementary transformations:
where \(\lambda =(5r_4+1)(5r_4-5r_1-5r_2+2)\). Since \(1\le r_1<r_2<r_3<p\), we can verfy that the matrix B is nonsingular, thus \(a_1=a_2=a_3=b_1=0 \), which contradicts with that \(b_1, a_j\in {\mathbb {F}}_p^*, 0\le j\le 3\).
Suppose that \(k=4\), that is
similarly, by \(c(1)=0\) and \(c^{(1)}(1)=c^{(1)}(\zeta )=c^{(2)}(1)=c^{(3)}(1)=0\), then we derive a contradiction.
Hence if \(c(x)=(V_i(x^5), V_j(x^5))\) with \((N_i, N_j)=(4,2)\), then \(w_p(c(x))=12\).
Case 2: If \(c(x)=(V_0(x^5), V_k(x^5)), 1\le k\le 4\), with \((N_0, N_k)=(3,3)\).
Let \(V_0(x^5)=a_0+ a_1x^{5r_1}+a_2x^{5r_2} \) with \(1\le r_1<r_2<p \) and \(V_k(x^5)=b_0+ b_1x^{5r_3}+b_2x^{5r_4}\) with \(1\le r_3<r_4 <p\), where \(a_0=-a_1-a_2\) and \(b_0=-b_1-b_2\). Then
It is obvious that if \(k\in \{2,3\}\), then \(L=6\) and \(w_p(c(x))=12\).
Suppose that \(k=1\). Then
by \(c^{(1)}(1)=c^{(1)}(\zeta )=c^{(2)}(1)=c^{(3)}(1)=0\), we have
where \(B_1,B_2,B_4\) are defined as (3.6), (3.7) and \(B_3^\prime \) is given by changing \(r_4\) into \(r_3\) in \(B_4\). Note that \(1\le r_1<r_2\) and \(1\le r_3<r_4\), we can obtain that the determinant of \(B=(B_1,B_2,B_3^\prime ,B_4)\) is not equal to 0. Hence \(k=1\) is impossible.
Suppose that \(k=4\), then
similar to the argument with \(k=1\), we know that \(k=4\) is also impossible.
Case 3: If \(c(x)= (V_0(x^5), V_i(x^5), V_j(x^5))\) with \((N_0, N_i, N_j)=(2,2,2)\) and \(1\le i<j\le 4\).
Let \(V_0(x^5)=a_1(x^{5r_1}-1)\), \(V_i(x^5)=a_2(x^{5r_2}-1)\), and \(V_j(x^5)=a_3(x^{5r_3}-1)\). Then
Note that \(1\le i<j\le 4\), then
The first and the second formal derivative of c(x) respectively gives
and
(1) Suppose that \((i,j)=(1,2)\). Since \((x -1)^5\) and \((x-\zeta )^2\) are divisors of c(x), \(c^{(1)}(1)=c^{(1)}(\zeta )=c^{(2)}(1)=0\). Then
where
We make some elementary transformations:
Note that \(1\le r_1,r_2,r_3<p\) are positive integers, we conclude that
The solution of Eq. (3.8) has only zero, which is a contradiction.
(2) Suppose that \((i,j)=(1,3)\). By \(c^{(1)}(1)=c^{(1)}(\zeta )=c^{(2)}(1)=0\), then
where \(B_1,B_2\) are defined as (3.9) and
(3) Suppose that \((i,j)=(1,4)\). By \(c^{(1)}(1)=c^{(1)}(\zeta )=c^{(2)}(1)=0\), then
where \(B_1,B_2\) are defined as (3.9) and
(4) Suppose that \((i,j)=(2,3)\). By \(c^{(1)}(1)=c^{(1)}(\zeta )=c^{(2)}(1)=0\), then
where \(B_1,B_3^\prime \) is defined as (3.9), (3.11), respectively, and
(5) Suppose that \((i,j)=(2,4)\). By \(c^{(1)}(1)=c^{(1)}(\zeta )=c^{(2)}(1)=0\), then
where \(B_1\), \(B_2^{\prime }\), and \( B_4 \) is defined as (3.9), (3.15), and (3.13), respectively.
(6) Suppose that \((i,j)=(3,4)\). By \(c^{(1)}(1)=c^{(1)}(\zeta )=c^{(2)}(1)=0\), then
where \(B_1\) and \( B_4 \) is defined as (3.9) and (3.13), respectively, \(B_2^{\prime \prime }\) is replaced \(r_3\) by \(r_2\) in \(B_3^\prime \) defined as (3.11).
Similar to the case \(i=1\) and \(j=2\), the solutions of (3.10), (3.12), (3.14), (3.16) and (3.17) are zero, which are contradictions.
Hence if \(w_H( c(x))=6\), then \(w_p(c(x))=12\). \(\square \)
Lemma 3.4
If \( w_H(c(x))=7\), then \(w_p(c(x))\ge 12\).
Proof
We divide into three cases to investigate \(w_p(c(x))\) with \( w_H(c(x))=7\).
Case 1: If \(c(x)=(V_0(x^5), V_k(x^5)), 1\le k\le 4\), with \((N_0, N_k)=(4,3)\).
Let \(V_0(x^5)=a_0+ a_1x^{5r_1}+a_2x^{5r_2}+a_3x^{5r_3}\) with \(1\le r_1<r_2<r_3<p\) and \(V_k(x^5)=b_0+ b_1x^{5r_4}+b_2x^{5r_5}\) with \(1\le r_4<r_5<p\). Then
It is obvious that if \(k\in \{2,3\}\), then \(L=7\) and \(w_p(c(x))=14\).
Suppose that \(k=1\). Then
then \(w_p(c(x))\ge 12\) except \( (r_4,r_5)\in \{ (r_1,r_2), (r_1,r_3), (r_2,r_3)\}\). Without loss of generality, we assume that \(r_4=r_1\) and \(r_5=r_2\). That is
in this case \(L=4\) and \(w_p(c(x))=11\). But, this is impossible. The details are the below.
The i-th \(1\le i\le 4\), formal derivative of c(x) respectively gives
and
Since \((x -1)^5\) and \((x-\zeta )^2\) are divisors of c(x), it follows from \(c^{(1)}(1)=c^{(1)}(\zeta )=c^{(2)}(1)=c^{(3)}(1)=c^{(4)}(1)=0\), note that \(a_0=-(a_1+a_2+a_3)\) and \(b_0=-(b_1+b_2)\), that
where B is defined as (3.5), \(\alpha =( r_2, r_2, r_2(5r_2+1), r_2(25r_2^2-1))^\top \), \(\beta = (r_1(5r_1-1)(5r_1-2)(5r_1-3),r_2(5r_2-1)(5r_2-2)(5r_2-3),r_3(5r_3-1)(5r_3-2)(5r_3-3),r_1(25r_1^2-1)(5r_1-2))\), and \( a_{55}=r_2(25r_2^2-1)(5r_2-2)\). By make some elementary transformations, note that \(1\le r_1<r_2<r_3<p\) and \( 1\le r_4<r_5<p\), we can check that the matrix \(\left( \begin{array}{cc} B&{} \alpha \\ \beta &{} a_{55}\end{array}\right) \) is nonsingular, hence the solution of (3.18) is zero, which is a contradiction.
Suppose that \(k=4\). Then
then \(w_p(c(x))\ge 12\) except \(r_1=1\) and \( (r_4,r_5)=(r_2-1,r_3-1)\). That is
using arguments similar to the above, \(c^{(1)}(1)=c^{(1)}(\zeta )=c^{(2)}(1)=c^{(3)}(1)=c^{(4)}(1)=0\), we derive a contradiction.
Hence if \(c(x)=( V_0(x^5), V_k(x^5)), 1\le k\le 4 \) with \((N_0, N_k)=(4,3)\), then \(w_p(c(x))\ge 12\).
Case 2: If \(c(x)=(V_0(x^5),V_k(x^5)), 1\le k\le 4\), with \((N_0, N_k)=(5,2)\). It is easy to see that \(L\ge 5\) and \(w_p(c(x))\ge 12\).
Case 3: If \(c(x)= (V_0(x^5), V_i(x^5), V_j(x^5))\) with \((N_0, N_i, N_j)=(2,2,3)\) and \( 1\le i<j\le 4\).
Let \(V_0(x^5)=a_1(x^{5r_1}-1)\), \(V_i(x^5)=a_2(x^{5r_2}-1)\), and \(V_j(x^5)= b_0+b_1x^{5r_3}+b_2x^{5r_4}\), where \(1\le r_3<r_4<p\). Then
Note that \(1\le i<j\le 4\), it is easy to check that \(w_p(c(x))\ge 12\) except
In the following, we discuss the subcases: (1) \(i=1\) and \(j=2\); (2) \( i=1 \) and \( j=4\); (3) \(i=3\) and \(j=4\).
The first, the second, and the third formal derivative of c(x) respectively gives
(1) Suppose that \((i,j)=(1,2)\). Since \((x -1)^5\) and \((x-\zeta )^2\) are divisors of c(x), \(c^{(1)}(1)=c^{(1)}(\zeta )=c^{(2)}(1)=c^{(3)}(1)=0\). Then
where
and
Note that \(r_1,r_2,r_3<r_4<p\) are positive integers and \(\zeta \) is a primitive 5-th root of unity in \({\mathbb {F}}_p\), by making some elementary transformations, we obtain \((B_1,B_2,B_3,B_4)\) is nonsingular. The solution of Eq. (3.19) has only zero, which is a contradiction.
(2) Suppose that \((i,j)=(1,4)\). By \(c^{(1)}(1)=c^{(1)}(\zeta )=c^{(2)}(1)=c^{(3)}(1)=0\), then
where \(B_1,B_2\) are defined as (3.20) and
(3) Suppose that \((i,j)=(3,4)\). By \(c^{(1)}(1)=c^{(1)}(\zeta )=c^{(2)}(1)=c^{(3)}(1)=0\), then
where \(B_1\) and \(B_3^\prime , B_4^\prime \) is defined as (3.20) and (3.23), respectively, and
Similar to the case \(i=1\) and \(j=2\), the solutions of (3.22) and (3.24) have zero, a contradiction.
Hence if \(w_H( c(x))=7\), then \(w_p(c(x))\ge 12\). \(\square \)
Lemma 3.5
If \( w_H(c(x))=8\), then \(w_p(c(x))\ge 12\).
Proof
If \( w(c(x))=8\). Suppose that \(c(x)= (V_0(x^5),V_k(x^5)),1\le k\le 4,\) with \((N_0, N_k)\in \{(2,6),(3,5),(4,4)\}\). Then \(w_p(c(x))\ge 12\). We only need to consider the following two cases.
Case 1: If \(c(x)= (V_0(x^5),V_i(x^5),V_j(x^5))\) with \((N_0, N_i,N_j)=(3, 3, 2)\) and \(1\le i<j\le 4\).
Let \(V_0(x^5)=a_0+ a_1x^{5r_1}+a_2x^{5r_2} \) with \(1\le r_1<r_2<p \), \(V_i(x^5)=b_0+ b_1x^{5r_3}+b_2x^{5r_4}\) with \(1\le r_3<r_4 <p\), and \(V_j(x^5)=b_3(x^{5r_5}-1)\), where \(a_0=-a_1-a_2\) and \(b_0=-b_1-b_2\). Then
Note that \(1\le i<j \le 4\), then
We can quickly check that \(d_p(c(x))\ge 12 \) except \( (i,j)\in \{ (1,2), (1,4)\}\).
(1) Suppose that \((i,j)=(1,2)\). We can now see that if \(r_1=r_3\) and \(r_2=r_4\), then \(w_p(c(x))=8+3=11\); otherwise, \(w_p(c(x))\ge 12\). Without loss of generality, we assume that \(r_1=r_3=1\) and \(r_2=r_4=2\). Then
By \(c^{(1)}(1) =c^{(1)}(\zeta )=c^{(2)}(1)=c^{(3)}(1)=c^{(4)}(1)=0\), note that \(a_0=-(a_1+a_2)\) and \(b_0=-(b_1+b_2)\), we have
it is easy to verify the solution of the above equation is zero, which is a contradiction.
(2) Suppose that \((i,j)=(1,4)\). We can easily observe that \(w_p(c(x))\ge 12\) except the case \(r_1=r_3=1\) and \(r_2=r_4=2\). In a similar way, by \(c^{(1)}(1) =c^{(1)}(\zeta )=c^{(2)}(1)=c^{(3)}(1)=c^{(4)}(1)=0\), this is also a contradiction.
Case 2: If \(c(x)= (V_0(x^5),V_i(x^5),V_j(x^5),V_k(x^5))\) with \((N_0, N_i,N_j,N_k)=(2,2,2,2)\) and \(1\le i<j<k\le 4\).
Let \(V_0(x^5)=a_1(x^{5r_1}-1)\), \(V_i(x^5)=a_2(x^{5r_2}-1)\), \(V_j(x^5)=a_3(x^{5r_3}-1)\), and \(V_k(x^5)=a_4(x^{5r_4}-1)\). Then
Note that \(1\le i<j<k\le 4\). Then
(1) Suppose that \((i,j,k)=(1,2,4)\). It follows from (3.25) that \(w_p(c(x))\ge 12\) except \(r_1=r_2=r_3=1\).
If \((i,j,k)=(1,2,4)\) and \(r_1=r_2=r_3=1\), then
By \(c^{(1)}(1) =c^{(1)}(\zeta )=c^{(2)}(1)=c^{(3)}(1)=0\), we have
where \(\mu =r_4((5r_4+4)(5r_4+5)+6)\). The solution of the above equation has only zero, which is a contradiction.
(2) Suppose that \((i,j,k)=(1,3,4)\). It follows from (3.25) that \(w_p(c(x))\ge 12\) except \(r_1=r_2=1\) and \(r_3=r_4<p\).
If \((i,j,k)=(1,3,4)\), \(r_1=r_2= 1\), and \(r_3=r_4\), then
A similar argument to the above, by \(c^{(1)}(1) =c^{(1)}(\zeta )=c^{(2)}(1)=c^{(3)}(1)=0\), we conclude that if \((i,j,k)=(1,3,4)\) and \(r_1=r_2=1\) and \(r_3=r_4<p\) is impossible.
(3) Suppose that \((i,j,k)=(2,3,4)\). It follows from (3.25) that \(w_p(c(x))\ge 12\) except \(r_1=1\) and \(r_2=r_3=r_4<p\).
If \((i,j,k)=(2,3,4)\), \(r_1= 1\), and \(r_2=r_3=r_4\), then
A similar argument to the above, by \(c^{(1)}(1) =c^{(1)}(\zeta )=c^{(2)}(1)=c^{(3)}(1)=0\), we know that if \((i,j,k)=(2,3,4)\) and \(r_1 =1\) and \(r_2=r_3=r_4\) is impossible.
(4) Suppose that \((i,j,k)=(1,2,3)\). It follows from (3.25) that \(w_p(c(x))\ge 12\) except \(r_1=r_2=r_3<p\) or \(r_2=r_3=r_4<p\) or \(r_1=r_2<p, r_3=r_4<p\). But the three cases are not happen.
Hence if \(w_H( c(x))=8\) then \(w_p(c(x))\ge 12\). \(\square \)
Now we are ready to complete the proof of Theorem 3.2.
Proof
From Lemmas 3.3, 3.4, 3.5, we know that for \(0\ne c(x)\in {\mathcal {C}}\), if \(6\le w_H(c(x))\le 8\), then \(w_p(c(x))\ge 12 \). Furthermore, if \(w_H(c(x))\ge 9\), then by (2.1), it is easy to verify that no such codeword c(x) in \({\mathcal {C}}\) exists such that \(w_p(c(x))< 12\). Hence we conclude that \(d_p({\mathcal {C}})=12\).
Therefore, if \(g(x)=(x-1)^5(x-\zeta )^2(x-\zeta ^2)(x-\zeta ^3)(x-\zeta ^4)\), then \({\mathcal {C}}=\langle g(x)\rangle \) is a (5p, 12) MDS symbol-pair code. This completes the proof of Theorem 3.2. \(\square \)
Example 3.6
Let \(p=11\) and \(g(x)=(x-1)^5(x-3)^2(x-9)(x-5)(x-4)\). Then \({\mathcal {C}}=\langle g(x)\rangle \) is a [55, 45, 6] cyclic code. By Theorem 3.2, its minimum symbol-pair distance is 12. The code \({\mathcal {C}}\) is an MDS symbol-pair code.
Example 3.7
Let \(p=31\) and \(g(x)=(x-1)^5(x-4)^2(x-16)(x-2)(x-8)\). Then \({\mathcal {C}}=\langle g(x)\rangle \) is a [155, 45, 6] cyclic code. By Theorem 3.2, its minimum symbol-pair distance is 12. The code \({\mathcal {C}}\) is an MDS symbol-pair code.
Suppose that \( 3|(p-1)\). Let
be a set of nontrivial cyclic codes of length 3p over \({\mathbb {F}}_p\), where \(\omega \) is a primitive 3-th root of unity in \({\mathbb {F}}_p\). From the proof of Theorem 3.1 and the results in [15], we have the following results.
Theorem 3.8
Let \({\mathcal {C}}=\langle g(x)\rangle \in S^\prime \) and \(d_H({\mathcal {C}})= 5\). Then there is a unique MDS symbol-pair code of length 3p over \({\mathbb {F}}_p\) as follows:
Let \({\mathcal {C}}=\langle g(x)\rangle \in S^\prime \) and \(d_H({\mathcal {C}})= 6\). Then there is a unique MDS symbol-pair code of length 3p over \({\mathbb {F}}_p\) as follows:
Furthermore, by the proof of Theorem 3.1, we know the following results.
Proposition 3.9
(1) If \({\mathcal {C}}=\langle g(x)\rangle \in S, d_H({\mathcal {C}})=8\), and \({\mathcal {C}}\) is an MDS symbol-pair code of length 5p over \({\mathbb {F}}_p\). Then there is a unique possible code as follows:
(2) If \({\mathcal {C}}=\langle g(x)\rangle \in S^\prime , d_H({\mathcal {C}})=7\), and \({\mathcal {C}}\) is an MDS symbol-pair code of length 3p over \({\mathbb {F}}_p\). Then there is a unique possible code as follows:
Question 3.10
In Proposition 3.9, are two codes MDS symbol-pair codes?
4 Concluding remarks
Let p be a prime and \(5|(p-1)\). Let S be a set of all repeated-root cyclic codes \({\mathcal {C}}=\langle g(x)\rangle \), \((x^5-1)|g(x)\), of length 5p over a field field \({\mathbb {F}}_p\). In this paper, we provided a method to find MDS symbol-pair codes in S whose Hamming distance is 6. By the method we can easily obtain the results in [15] and new MDS symbol-pair codes of length \(\ell p\) over \({\mathbb {F}}_p\), where \(\ell \) is a positive integer with \(\ell |(p-1)\) and \((x^{\ell }-1)|g(x)\).
References
Cassuto Y., Blaum M.: Codes for symbol-pair read channels. IEEE Trans. Inf. Theory 57(12), 8011–8020 (2011).
Castagnoli G., Massey J.L., Schoeller P.A., von Seemann N.: On repeated-root cyclic codes. IEEE Trans. Inf. Theory 37(2), 337–342 (1991).
Cassuto Y., Litsyn S.: Symbol-pair codes: algebraic constructions and asymptotic bounds. In: Proceedings of the IEEE International Symposium on Information Theory, Saint Petersburg, Russia, pp. 2348–2352 (2011)
Chee Y.M., Ji L., Kiah H.M., Wang C., Yin J.: Maximum distance separable codes for symbol-pair read channels. IEEE Trans. Inf. Theory 59(11), 7259–7267 (2013).
Chee Y.M., Kiah, H.M., Wang, C.: Maximum distance separable symbol-pair codes. In: Proceedings of IEEE International Symposium Information Theory (ISIT), pp. 2886–2890 (2012).
Chen B., Lin L., Liu H.: Constacyclic symbol-pair codes: lower bounds and optimal constructions. IEEE Trans. Inf. Theory 63(12), 7661–7666 (2017).
Ding B., Ge G., Zhang J., Zhang T., Zhang Y.: New constructions of MDS symbol-pair codes. Des. Codes Cryptogr. 86, 841–859 (2018).
Ding B., Zhang T., Ge G.: Maximum distance separable codes for \(b\)-symbol read channels. Finite Fields Appl. 49, 180–197 (2018).
Dinh H.Q., Nguyen B.T., Singh A.K., Sriboonchitta S.: On the symbol-pair distance of repeated-root constacyclic codes of prime power lengths. IEEE Trans. Inf. Theory 64(4), 2417–2430 (2018).
Dinh H.Q., Wang X., Liu H., Sriboonchitta S.: On the symbol-pair distance of repeated-root constacyclic codes of length \(2p^s\). Discret. Math. 342(11), 3062–3078 (2019).
Dinh H.Q., Wang X., Liu H., Sriboonchitta S.: On the \(b\)-distance of repeated-root constacyclic codes of prime power lengths. Discret. Math. 343(4), 111780 (2020).
Kai X., Zhu S., Li P.: A construction of new MDS symbol-pair codes. IEEE Trans. Inf. Theory 61(11), 5828–5834 (2015).
Kai X., Zhu S., Zhao Y., Luo H., Chen Z.: New MDS symbol-pair codes from repeated-root codes. IEEE Commun. Lett. 22(3), 462–465 (2018).
Li S., Ge G.: Constructions of maximum distance separable symbol-pair codes using cyclic and constacyclic codes. Des. Codes Cryptogr. 84(3), 359–372 (2017).
Ma J., Luo J.: MDS symbol-pair codes from repeated-root cyclic codes. Des. Codes Cryptogr. 90, 121–137 (2022).
Yaakobi E., Bruck J., Siegel P.H.: Constructions and decoding of cyclic codes over \(b\)-symbol read channels. IEEE Trans. Inf. Theory 62(4), 1541–1551 (2016).
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The paper was supported by National Natural Science Foundation of China under Grant 12171420, Natural Science Foundation of Shandong Province under Grant ZR2021MA046, and Plateau science of Zaozhuang University.
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Li, F. Cyclic codes of length 5p with MDS symbol-pair. Des. Codes Cryptogr. 91, 1873–1888 (2023). https://doi.org/10.1007/s10623-023-01184-x
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DOI: https://doi.org/10.1007/s10623-023-01184-x