Abstract
Symbol-pair code is a new coding framework which is proposed to correct errors in the symbol-pair read channel. In particular, maximum distance separable (MDS) symbol-pair codes are a kind of symbol-pair codes with the best possible error-correction capability. Employing cyclic and constacyclic codes, we construct three new classes of MDS symbol-pair codes with minimum pair-distance five or six. Moreover, we find a necessary and sufficient condition which ensures a class of cyclic codes to be MDS symbol-pair codes. This condition is related to certain property of a special kind of linear fractional transformations. A detailed analysis on these linear fractional transformations leads to an algorithm, which produces many MDS symbol-pair codes with minimum pair-distance seven.
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Acknowledgments
The authors wish to thank the anonymous reviewers for their comments which are very helpful to improve the paper. The first author would like to express his gratitude to Prof. Maosheng Xiong, Hong Kong University of Science and Technology, for the enlightening discussions on this topic. The research of G. Ge is supported by the National Natural Science Foundation of China under Grant Nos. 11431003 and 61571310.
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Communicated by C. Mitchell.
Appendix
Appendix
Let q be a prime power. For \(u,v,w,z \in \mathbb {F}_q\) and \(\delta \in \mathbb {F}_{q^2}\), define a linear fractional transformation from \(\mathbb {F}_{q^2}\) to \(\mathbb {F}_{q^2}\) by
where \(w+z\delta \ne 0\) and \(uz-vw \ne 0\). We further assume that \(z \ne 0\), since otherwise, \(f_{u,v,w,z}\) degenerates into a linear function. Below, we will study this special kind of linear fractional transformation. In particular, suppose \(\delta \in \mathbb {F}_{q^2}{\setminus } \mathbb {F}_q\), we will present a necessary and sufficient condition such that
for some integer i. This condition provides a criterion to determine whether the linear fractional transformation \(f_{u,v,w,z}\) maps \(\delta \) to an element belonging to the multiplicative cyclic group generated by \(\delta \).
Proposition 17
Let \(\delta \in \mathbb {F}_{q^2}{\setminus } \mathbb {F}_q\). Let \(x^2-bx-c\) be the monic minimal polynomial of \(\delta \) over \(\mathbb {F}_q\). For an integer \(i \ge 2\), define
Then for \(i \ge 0\), \(\delta ^i=\frac{u+v\delta }{w+z\delta }\) if and only if one of the following holds:
-
(1)
If \(i=0\), then \(u=w\), \(v=z\).
-
(2)
If \(i=1\), then \(b=\frac{v-w}{z}\) and \(c=\frac{u}{z}\).
-
(3)
If \(i \ge 2\), then
$$\begin{aligned} a_1^{(i)}\ne 0, \quad b=-\frac{a_0^{(i)}}{a_1^{(i)}}+\frac{v}{za_1^{(i)}}-\frac{w}{z}, \quad c=-\frac{wa_0^{(i)}}{za_1^{(i)}}+\frac{u}{za_1^{(i)}}. \end{aligned}$$
Proof
(1) and (2) are trivial. We only consider (3) below. Since \(\delta ^i=\frac{u+v\delta }{w+z\delta }\) and \(z \ne 0\), we have \(\delta ^{i+1}+\frac{w}{z}\delta ^{i}-\frac{v}{z}\delta -\frac{u}{z}=0\). Therefore, \(\delta \) is a root of the polynomial \(x^{i+1}+\frac{w}{z}x^{i}-\frac{v}{z}x-\frac{u}{z}\) and
For an integer \(i\ge 0\), we define a polynomial \(T_i(x)=x^{i+1}+\frac{w}{z}x^{i}\). For any \(i \ge 2\), we have the following recurrence relation:
By employing this recurrence relation repeatedly, we have
where \(d_1^{(i)},d_2^{(i)},e_0^{(i)},e_1^{(i)}\in \mathbb {F}_q\). Now, we aim to determine \(e_0^{(i)}\) and \(e_1^{(i)}\) explicitly. The recurrence relation implies that \(T_0(x)\) necessarily originates from \(T_2(x)\) by subtracting a proper multiple of \(x^2-bx-c\). Since \(T_2(x)\equiv bT_{1}(x)+cT_{0}(x) \pmod {x^2-bx-c}\), we have \(e_0^{(i)}=c d_2^{(i)}\). Apparently, \(d_2^{(i)}\) is a summation of monomials regarding of b and c. More precisely, suppose \(i-2\) can be expressed as an ordered sum containing \(i-2-2j\) ones and j twos. Then this ordered sum corresponds to a monomial \(b^{i-2-2j}c^j\) in the summation of \(d_2^{(i)}\). Recall that there are \(\left( {\begin{array}{c}i-2-j\\ j\end{array}}\right) \) ways to decompose \(i-2\) into distinct ordered sums containing \(i-2-2j\) ones and j twos. Therefore, we have
and
Similarly, by analyzing the decomposition of \(i-1\) into ordered sums consisting of ones and twos, we have
Consequently,
Hence, we must have \(a_1^{(i)}\ne 0\) and \(x^2+(\frac{a_0^{(i)}}{a_1^{(i)}}+\frac{w}{z}-\frac{v}{za_1^{(i)}})x+\frac{wa_0^{(i)}}{za_1^{(i)}}-\frac{u}{za_1^{(i)}}=x^2-bx-c\). The conclusion follows by comparing the coefficients. \(\square \)
Particularly, given \(\delta \in \mathbb {F}_{q^2}{\setminus } \mathbb {F}_q\) and an integer \(i\ge 2\), we have the following easy criterion to determine if \(\delta ^i=\frac{1-\theta \delta }{-\theta +\delta }\) for some \(\theta \in \mathbb {F}_q^*\).
Corollary 18
Let \(\delta \in \mathbb {F}_{q^2}{\setminus } \mathbb {F}_q\). Let \(x^2-bx-c\) be the monic minimal polynomial of \(\delta \) over \(\mathbb {F}_q\). For an integer \(i \ge 2\), \(\delta ^i=\frac{1-\theta \delta }{-\theta +\delta }\) for some \(\theta \in \mathbb {F}_q^*\) if and only if \(a_1^{(i)}\ne 0\) and one of the following condition holds
-
1)
If \(a_1^{(i)}=1\), then \(a_0^{(i)}=-b\) and \(c \ne 1\),
-
2)
If \(a_0^{(i)}=0\), then \(a_1^{(i)}=\frac{1}{c}\) and \(b \ne 0\),
-
3)
If \(a_0^{(i)}\ne 0\) and \(a_1^{(i)}\ne 1\), then \(a_1^{(i)}c\ne 1\) and \(\frac{a_1^{(i)}b+a_0^{(i)}}{a_1^{(i)}-1}=\frac{a_1^{(i)}c-1}{a_0^{(i)}}\).
where \(a_0^{(i)}\) and \(a_1^{(i)}\) are defined in (4). Moreover, let \(\mathbb {F}_r\) be a subfield of \(\mathbb {F}_q\). If \(b,c \in \mathbb {F}_r\), then \(\delta ^i=\frac{1-\theta \delta }{-\theta +\delta }\) for some \(i\ge 2\) only if \(\theta \in \mathbb {F}_r\).
Proof
By setting \(u=z=-1\) and \(v=w=\theta \) in Proposition 17, we have \(\delta ^i=\frac{1-\theta \delta }{-\theta +\delta }\) for some \(\theta \in \mathbb {F}_q^*\) if and only if
If \(a_0^{(i)}=0\) and \(a_1^{(i)}=1\), then we have \(b=0\) and \(c=1\), which is impossible since \(x^2-1\) is reducible over \(\mathbb {F}_q\). If either \(a_1^{(i)}=1\) or \(a_0^{(i)}=0\), then the Condition 1) or the Condition 2) holds. If \(a_0^{(i)}\ne 0\) and \(a_1^{(i)}\ne 1\), the Condition 3) is derived from the expressions of b and c. Suppose b and c belong to a subfield \(\mathbb {F}_r\), then \(a_0^{(i)},a_1^{(i)}\in \mathbb {F}_r\) by definition. Since we have either \(a_0^{(i)}\ne 0\) or \(a_1^{(i)}\ne 1\), it is easy to see that \(\theta \in \mathbb {F}_r\). \(\square \)
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Li, S., Ge, G. Constructions of maximum distance separable symbol-pair codes using cyclic and constacyclic codes. Des. Codes Cryptogr. 84, 359–372 (2017). https://doi.org/10.1007/s10623-016-0271-y
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DOI: https://doi.org/10.1007/s10623-016-0271-y
Keywords
- Algebraic construction
- Constacyclic codes
- Cyclic codes
- Linear fractional transformations
- MDS symbol-pair codes
- Symbol-pair codes