Abstract
Recently, linear codes constructed from defining sets have been studied widely and they have many applications. For an odd prime p, let \(q=p^{m}\) for a positive integer m and \(\mathrm {Tr}_{m}\) the trace function from \(\mathbb {F}_{q}\) onto \(\mathbb {F}_{p}\). In this paper, for a positive integer t, let \(D\subset \mathbb {F}^{t}_{q}\) and \(D=\{(x_{1},x_{2}) \in (\mathbb {F}_{q}^{*})^{2} : \mathrm {Tr}_{m}(x_{1}+x_{2})=0\}\), we define a p-ary linear code \(\mathcal {C}_{D}\) by
where
We compute the weight enumerators of the punctured codes \(\mathcal {C}_{D}\).
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1 Introduction
Let \(\mathbb {F}_{p}\) be the finite field with p elements, where p is an odd prime. An [n, k, d] linear code \(\mathcal {C}\) over \(\mathbb {F}_{p}\) is a k-dimensional subspace of \(\mathbb {F}^{n}_{p}\) with minimum distance d. Let \(A_{i}\) denote the number of codewords with Hamming weight i the code \(\mathcal {C}\) of length n. The weight enumerator of \(\mathcal {C}\) is defined by \(1+A_{1}z+A_{2}z^{2}+\cdots +A_{n}z^{n}\). The sequence \((1,A_{1},A_{2},\ldots ,A_{n})\) is called the weight distribution of the code \(\mathcal {C}\). The weight distribution of the linear code is an important subject in coding theory. However, it is difficult to compute the weight distribution of a linear code in general.
Recently, the weight enumerators of linear codes were studied in [1, 2, 4,5,6, 9,10,11,12, 15,16,18] with the help of exponential sums in some cases. Ahn, Ka and Li [1] defined a class of linear codes as follows. Let \(D' = \{(x_{1},x_{2},\ldots ,x_{t}) \in \mathbb {F}_{q}^{t}{\setminus } \{(0,0,\ldots ,0)\} : \mathrm {Tr}_{m}(x_{1}+x_{2}+\cdots +x_{t})=0\}.\) A p-ary linear code \(\mathcal {C}_{D'}\) is defined by
where
They determined the complete weight enumerators of \(\mathcal {C}_{D'}\). Yang and Yao [17] generalized the results of Ahn, Ka and Li [1]. They defined \(D_{b} = \{(x_{1},x_{2},\ldots ,x_{t}) \in \mathbb {F}_{q}^{t} : \mathrm {Tr}_{m}(x_{1}+x_{2}+\cdots +x_{t})=b\}\) for any \(b \in \mathbb {F}_{p}^{*}\) and determined the complete weight enumerator of a class of p-ary linear codes given by
where
In this paper, we define
and a p-ary linear code \(\mathcal {C}_{D}\) by
where
The purpose of this paper is to compute the weight enumerators of the punctured codes \(\mathcal {C}_{D}\).
Minimal linear codes can be used to construct secret sharing schemes with interesting access structures [7, 8]. The codes presented in this paper are minimal in the sense of Ding and Yuan [7, 8]. We shall explain it at the end of this paper in detail.
2 Preliminaries
Let p be an odd prime and \(q=p^{m}\) for a positive integer m. For any \(a\in \mathbb {F}_{q}\), we can define an additive character of the finite field \(\mathbb {F}_{q}\) as follows:
where \(\zeta _p=e^{\frac{2\pi \sqrt{-1}}{p}}\) is a p-th primitive root of unity and \(\mathrm {Tr}_{m}\) denotes the trace function from \(\mathbb {F}_{q}\) onto \(\mathbb {F}_{p}\). It is clear that \(\psi _0(x)=1\) for all \(x \in \mathbb {F}_{q}\). Then \(\psi _0\) is called the trivial additive character of \(\mathbb {F}_{q}\). If \(a=1\), we call \(\psi :=\psi _1\) the canonical additive character of \(\mathbb {F}_{q}\). It is easy to see that \(\psi _a(x)=\psi (ax)\) for all \(a, x \in \mathbb F_q\). The orthogonal property of additive characters is given by
Let \(\lambda : \mathbb F_q^* \rightarrow \mathbb C^*\) be a multiplicative character of \(\mathbb F_q^*\). Now we define the Gauss sum over \(\mathbb {F}_{q}\) by
Let \(q-1=sN\) for two positive integers \(s>1\), \(N>1\) and \(\alpha \) be a fixed primitive element of \(\mathbb {F}_{q}\). Let \(\langle \alpha ^{N} \rangle \) denote the subgroup of \(\mathbb {F}_{q}^{*}\) generated by \(\alpha ^{N}\). The cyclotomic classes of order N in \(\mathbb {F}_{q}\) are the cosets \(C_{i}^{(N,q)}=\alpha ^{i}\langle \alpha ^{N}\rangle \) for \(i=0,1,\ldots , N-1.\) We know that \(|C_{i}^{(N,q)}|=\frac{q-1}{N}\). The Gaussian periods of order N are defined by
Suppose that \(\eta \) is the quadratic character of \(\mathbb {F}^{*}_{q}\) and \(\eta _{p}\) is the quadratic character of \(\mathbb {F}^{*}_{p}.\) For \(z\in \mathbb {F}^{*}_{p}\), it is easily checked that
Lemma 1
[3, 13] Suppose that \(q=p^m\) where p is an odd prime and \(m \ge 1\). Then
where \(p^*=\big (\frac{-1}{p}\big )p=(-1)^{\frac{p-1}{2}}p\).
Lemma 2
[13] If q is odd and \(f(x)=a_{2}x^{2}+a_{1}x+a_{0}\in \mathbb {F}_{q}[x]\) with \(a_{2}\ne 0,\) then
Lemma 3
[14] When \(N=2\), the Gaussian periods are given by
and \(\eta ^{(2,q)}_{1}=-1-\eta ^{(2,q)}_{0}\).
3 Weight enumerators of the linear codes of \(\mathcal {C}_{D}\)
In this section, we present the weight distribution of the linear code \(\mathcal {C}_{D}\) defined by (1) and (2), where
To get the length of \(\mathcal {C}_{D}\), we need the following lemma.
Lemma 4
Denote \(n_{c}=|\{x_{1}, x_{2}, \in \mathbb {F}_{q}^{*} :\mathrm {Tr}_{m}(x_{1}+x_{2})=c\}|\) for each \(c \in \mathbb {F}_{p}\). Then
Proof
By the orthogonal property of additive characters, we have
Thus, we get the desired results. \(\square \)
By Lemma 4 it is easy to see that the length of \(\mathcal {C}_{D}\) is \(n_{0}=\frac{(p^{m}-1)^{2}+p-1}{p}\).
For a codeword \(\mathbf {c}(a_{1},a_{2})\) of \(\mathcal {C}_{D}\) and \(\rho \in \mathbb {F}_{p}^{*},\) let \(N_{0}:=N(a_{1},a_{2})\) be the number of components \(\mathrm {Tr}_{m}(a_{1}x^{2}_{1}+a_{2}x^{2}_{2})\) of \(\mathbf {c}(a_{1},a_{2})\) which are equal to 0. Then
where
We are going to determine the values of \(\varOmega _{2}\) and \(\varOmega _{3}\) in Lemmas 5 and 6. To simplify formulas, denote \(G_{i}=G(\eta )\eta (a_{i})\) for \(i\in \{1,2\}\).
Lemma 5
If \(a_{1}=0\) and \(a_{2}=0\), then
(1) If m is even, then
(2) If m is odd, then
Proof
When \(a_{1}=0\) and \(a_{2}=0\), it is obvious that \(\varOmega _{2}\) is equal to \((p^{m}-1)^{2}(p-1)\).
If \(a_{1}\ne 0\) and \(a_{2}=0\), then by the orthogonal property of additive characters, we have
By Lemma 2, we obtain
By (3), we get the results. Similarly, we compute the value of \(\varOmega _{2}\) when \(a_{1}=0\) and \(a_{2}\ne 0\).
If \(a_{1}\ne 0\) and \(a_{2}\ne 0\), then by Lemma 2, we get
By (3), we get the results. \(\square \)
To simplify results, we denote \(G(\eta )G(\eta _{p})\) by G and \(A_{i}=\eta (a_{i})\eta _{p}(-\mathrm {Tr}_{m}(a_{i}^{-1}))\) for \(i \in \{1,2\}\).
Lemma 6
If \(a_{1}=0\) and \(a_{2}=0\), then
Suppose that m is even.
(1) If \(a_{1}\ne 0\) and \(a_{2}=0\), then
(2) If \(a_{1}=0\) and \(a_{2}\ne 0\), then
(3) If \(a_{1}\ne 0\) and \(a_{2}\ne 0\), then
Suppose that m is odd.
(1) If \(a_{1}\ne 0\) and \(a_{2}=0\), then
(2) If \(a_{1}=0\) and \(a_{2}\ne 0\), then
(3) If \(a_{1}\ne 0\) and \(a_{2}\ne 0\), then
Proof
We only compute the value of \(\varOmega _{3}\) for the case \(a_{1}\ne 0\) and \(a_{2}\ne 0\). One can compute the other cases similarly. By the orthogonal property of additive characters, we have
By Lemma 2, we obtain
If one of \(\mathrm {Tr}_{m}(a_{1}^{-1})\), \(\mathrm {Tr}_{m}(a_{2}^{-1})\) and \(\mathrm {Tr}_{m}(a_{1}^{-1}+a_{2}^{-1})\) is zero, then it is easy to compute the term corresponding to it. We only consider the case of \(\mathrm {Tr}_{m}(a_{1}^{-1})\ne 0\), \(\mathrm {Tr}_{m}(a_{2}^{-1})\ne 0\) and \(\mathrm {Tr}_{m}(a_{1}^{-1}+a_{2}^{-1})\ne 0\). Other cases can be computed similarly. From (5) we have
By Lemma 2, we obtain
By (3), we get the result. \(\square \)
Then by Lemmas 5 and 6, we obtain the values of \(N_{0}\). To get the frequency of each composition, we need the following lemmas.
Lemma 7
[1, Lemma 3.4] For any \(c \in \mathbb {F}_{p}\), let
Then we have
Lemma 8
[1, Lemma 3.5] For any \(c\in \mathbb {F}_{p}\), let
(1) If m is even, then
(2) If m is odd, then
Proof
If \(c=0\), then we get the result from [1, Lemma 3.5] with \(t=1\). If \(c\ne 0\), then by the orthogonal property of additive characters, we have
Assume that m is even, then 2 divides \(\frac{q-1}{p-1}\) and so \(\mathbb {F}^{*}_{p}\subseteq C_{0}^{(2,q)}.\) By (6) we obtain
Thus, we get the results. Also the case of \(n_{-1,c}\) is proved similarly.
Now suppose that m is odd, then \(\mathbb {F}^{*}_{p}=\{\mathbb {F}^{*}_{p}\cap C_{0}^{(2,q)}\}\cup \{\mathbb {F}^{*}_{p}\cap C_{1}^{(2,q)}\}.\) i.e.,\(|\mathbb {F}^{*}_{p}\cap C_{0}^{(2,q)}|=|\mathbb {F}^{*}_{p}\cap C_{1}^{(2,q)}|=\frac{p-1}{2}.\) By (6) we obtain
If \(-c \in C_{0}^{(2,p)}\), then we have
If \(-c \in C_{1}^{(2,p)}\), then we have
It is easily checked that \(\eta _{0}^{(2,p)}\eta _{0}^{(2,q)}+\eta _{1}^{(2,p)}\eta _{1}^{(2,q)}=\frac{G+1}{2}\) and \(\eta _{1}^{(2,p)}\eta _{0}^{(2,q)}+\eta _{0}^{(2,p)}\eta _{1}^{(2,q)}=\frac{-G+1}{2}\). Thus, we get the results. Also \(n_{-1,c}\) is computed similarly. This completes the proof. \(\square \)
Lemma 9
[1, Lemma 3.7] Suppose that m is odd, let
Then we have
Theorem 1
Let \(\mathcal {C}_{D}\) be a linear code defined by (1) and (2) where \(D=\{(x_{1},x_{2}) \in (\mathbb {F}_{q}^{*})^{2} : \mathrm {Tr}_{m}(x_{1}+x_{2})=0\}.\) Suppose that m is even. If \(m=2\), then the weight distribution of \(\mathcal {C}_{D}\) is given by Table 1 and the code \(\mathcal {C}_{D}\) has parameters \([\frac{(p^{2}-1)^{2}+p-1}{p},\;4,\;(p-1)(p^{2}-p-2)]\). If \(m\ge 4\), then the weight distribution of \(\mathcal {C}_{D}\) is given by Table 2 and the code \(\mathcal {C}_{D}\) has parameters \([\frac{(p^{m}-1)^{2}+p-1}{p},\;2m,\;(p-1)\big (p^{2(m-1)}-p^{m-2}-p^{\frac{3m-4}{2}})]\).
Proof
Recall that \(N_{0}=\frac{(q-1)^{2}}{p^{2}}+\frac{1}{p^{2}}(\varOmega _{1}+\varOmega _{2}+\varOmega _{3})\). We employ Lemmas 5 and 6 to compute \(N_{0}\).
Assume that \(a_{1}\ne 0\) and \(a_{2}=0\).
If \(\mathrm {Tr}_{m}(a_{1}^{-1})=0\), then we obtain
Thus,
Now the frequencies are \(n_{1,0}\) and \(n_{-1,0}\) in Lemma 8, respectively.
If \(\mathrm {Tr}_{m}(a_{1}^{-1})\ne 0\), then we obtain
Thus,
Now the frequencies are \(\displaystyle \sum \nolimits _{c \in \mathbb {F}^{*}_{p}}n_{1,c}\) and \(\displaystyle \sum \nolimits _{c \in \mathbb {F}^{*}_{p}}n_{-1,c}\), respectively. If \(a_{1}=0\) and \(a_{2}\ne 0\), then we also have the same weights and the same frequencies with the case of \(a_{1}\ne 0\) and \(a_{2}=0\).
Now, assume that \(a_{1}\ne 0\) and \(a_{2}\ne 0\).
If \(\mathrm {Tr}_{m}(a_{1}^{-1})=\mathrm {Tr}_{m}(a_{2}^{-1})=0\), then we obtain
Thus,
Now the frequencies are \((n_{1,0})^{2}\), \(2n_{1,0}n_{-1,0}\), \((n_{-1,0})^{2}\), respectively.
If \(\mathrm {Tr}_{m}(a_{1}^{-1})\ne 0\;{\mathrm{and}}\;\mathrm {Tr}_{m}(a_{2}^{-1})=0\), then we have
Thus,
Now the frequencies are \(\displaystyle \left( \sum \nolimits _{c \in \mathbb {F}^{*}_{p}}m_{c}\right) n_{1,0}\) and \(\displaystyle \left( \sum \nolimits _{c \in \mathbb {F}^{*}_{p}}m_{c}\right) n_{-1,0}\), in Lemmas 7 and 8, respectively.
If \(\mathrm {Tr}_{m}(a_{1}^{-1})=0\;{\mathrm{and}}\;\mathrm {Tr}_{m}(a_{2}^{-1})\ne 0\), then we have the same weights and the same frequencies with the case of \(\mathrm {Tr}_{m}(a_{1}^{-1})\ne 0\;{\mathrm{and}}\;\mathrm {Tr}_{m}(a_{2}^{-1})=0\).
If \(\mathrm {Tr}_{m}(a_{1}^{-1})\ne 0,\mathrm {Tr}_{m}(a_{2}^{-1})\ne 0\;{\mathrm{and}}\;\mathrm {Tr}_{m}(a_{1}^{-1}+a_{2}^{-1})=0\), then we have
Thus,
Now the frequencies are \(\displaystyle \sum \nolimits _{c \in \mathbb {F}^{*}_{p}}n_{1,c}^{2}+n_{-1,c}^{2}\) and \(\displaystyle 2\sum \nolimits _{c \in \mathbb {F}^{*}_{p}}n_{1,c}n_{-1,c}\), respectively.
If \(\mathrm {Tr}_{m}(a_{1}^{-1})\ne 0,\mathrm {Tr}_{m}(a_{2}^{-1})\ne 0\;{\mathrm{and}}\;\mathrm {Tr}_{m}(a_{1}^{-1}+a_{2}^{-1})\ne 0,\) then we have
And the frequency is
It is equal to \(\displaystyle T-\sum \nolimits _{c \in \mathbb {F}^{*}_{p} }\big (m_{c}m_{-c}),\) where \(T=|\{(a_{1},a_{2}) \in (\mathbb {F}^{*}_{q})^{2} : \mathrm {Tr}_{m}(a_{1}^{-1})\ne 0 \quad {\mathrm{and}}\quad \mathrm {Tr}_{m}(a_{2}^{-1})\ne 0\}|.\) By Lemmas 4, 7 we get
Thus we compute the frequency.
Since the Hamming weight of \(\mathbf {c}(a_{1},a_{2})\) is equal to \(W_{H}(\mathbf {c}(a_{1},a_{2}))=n_{0}-N_{0}\), we immediately have the desired results. \(\square \)
Example 1
(1) Let \(p=3\) and \(m=2.\) Then \(q=9\) and \(n=22.\) By Theorem 1, the code \(\textit{C}_{D}\) is a [22, 4, 8] linear code. Its weight enumerator is
which is checked by Magma.
(2) Let \(p=5\) and \(m=2.\) Then \(q=25\) and \(n=116.\) By Theorem 1, the code \(\textit{C}_{D}\) is a [116, 4, 72] linear code. Its weight enumerator is
which is checked by Magma.
(3) Let \(p=3\) and \(m=4.\) Then \(q=81\) and \(n=2134.\) By Theorem 1, the code \(\textit{C}_{D}\) is a [2134, 8, 1278] linear code. Its weight enumerator is
which is checked by Magma.
Theorem 2
Let \(\mathcal {C}_{D}\) be a linear code defined by (1) and (2) where \(D=\{(x_{1},x_{2}) \in (\mathbb {F}_{q}^{*})^{2} : \mathrm {Tr}_{m}(x_{1}+x_{2})=0\}.\) Suppose that m is odd and \(m \ge 3\). Then the weight distribution of \(\mathcal {C}_{D}\) is given by Table 3 and the code \(\mathcal {C}_{D}\) has parameters \(\left[ \frac{(p^{m}-1)^{2}+p-1}{p},\;2m,\;(p-1)\big (p^{2(m-1)}-2p^{m-2}-p^{m-1}-2p^{\frac{m-3}{2}}) \right] \).
Proof
Recall that \(N_{0}=\frac{(q-1)^{2}}{p^{2}}+\frac{1}{p^{2}}(\varOmega _{1}+\varOmega _{2}+\varOmega _{3})\). We employ Lemmas 5 and 6 to compute \(N_{0}\).
Suppose that \(a_{1}\ne 0\) and \(a_{2}=0\).
If \(\mathrm {Tr}_{m}(a_{1}^{-1})=0\), then we obtain
Now the frequency is \(m_{0}\) in Lemma 7.
If \(\mathrm {Tr}_{m}(a_{1}^{-1})\ne 0\), then we obtain
Thus,
Now the frequencies are \(n^{\prime }_{1,1}+n^{\prime }_{-1,-1}\) and \(n^{\prime }_{1,-1}+n^{\prime }_{-1,1}\) in Lemma 9, respectively. If \(a_{1}=0\) and \(a_{2}\ne 0\), then we have the same values and the same frequencies with the case of \(a_{1}\ne 0\) and \(a_{2}=0\).
Now, assume that \(a_{1}\ne 0\) and \(a_{2}\ne 0\).
If \(\mathrm {Tr}_{m}(a_{1}^{-1})=\mathrm {Tr}_{m}(a_{2}^{-1})=0\), then we obtain
Thus,
Now the frequencies are \(n_{1,0}^{2}+n_{-1,0}^{2}\) and \(2n_{1,0}n_{-1,0}\) in Lemma 8, respectively.
If \(\mathrm {Tr}_{m}(a_{1}^{-1})\ne 0\) and \(\mathrm {Tr}_{m}(a_{2}^{-1})=0\), then we have
Thus,
Now the frequencies are \(m_{0}(n^{\prime }_{1,1}+n^{\prime }_{-1,-1})\) and \(m_{0}(n^{\prime }_{1,-1}+n^{\prime }_{-1,1})\), respectively. If \(\mathrm {Tr}_{m}(a_{1}^{-1})=0\) and \(\mathrm {Tr}_{m}(a_{2}^{-1})\ne 0\), then we have the same values and the same frequencies with the case of \(\mathrm {Tr}_{m}(a_{1}^{-1})\ne 0\) and \(\mathrm {Tr}_{m}(a_{2}^{-1})=0\).
If \(\mathrm {Tr}_{m}(a_{1}^{-1})\ne 0\), \(\mathrm {Tr}_{m}(a_{2}^{-1})\ne 0\) and \(\mathrm {Tr}_{m}(a_{1}^{-1}+a_{2}^{-1})=0\), then we have
Since \(\mathrm {Tr}_{m}(a_{1}^{-1}+a_{2}^{-1})=0\), we have \(A_{1}A_{2}=\eta (a_{1}a_{2})\eta _{p}(-1)\).
Assume that \(p \equiv 1 \pmod 4\). Then,
Now the frequencies are \(\displaystyle \sum \nolimits _{c \in C_{0}^{(2,p)}}n_{1,c}n_{1,-c}+\sum \nolimits _{c \in C_{1}^{(2,p)}}n_{-1,c}n_{-1,-c}\), \(\displaystyle \sum \nolimits _{c \in C_{1}^{(2,p)}}n_{1,c}n_{1,-c}+\sum \nolimits _{c \in C_{0}^{(2,p)}}n_{-1,c}n_{-1,-c}\),\(\displaystyle \sum \nolimits _{c \in C_{0}^{(2,p)}}n_{1,c}n_{-1,-c}+\sum \nolimits _{c \in C_{1}^{(2,p)}}n_{1,c}n_{-1,-c}+\sum \nolimits _{c \in C_{0}^{(2,p)}}n_{-1,c}n_{1,-c}+\sum \nolimits _{c \in C_{1}^{(2,p)}}n_{-1,c}n_{1,-c},\) in Lemma8, respectively.
In the case of \(p \equiv 3 \pmod 4\), we compute similarly.
If \(\mathrm {Tr}_{m}(a_{1}^{-1})\ne 0\), \(\mathrm {Tr}_{m}(a_{2}^{-1})\ne 0\) and \(\mathrm {Tr}_{m}(a_{1}^{-1}+a_{2}^{-1})\ne 0\), then we have
Thus,
We compute the frequency for the case of \(A_{1}A_{2}=-1\). We compute similarly for the other cases. From Lemma 8, the frequency is
\(\square \)
Example 2
(1) Let \(p=3\) and \(m=3.\) Then \(q=27\) and \(n=226.\) By Theorem 2, the code \(\textit{C}_{D}\) is a [226, 6, 128] linear code. Its weight enumerator is
which is checked by Magma.
(2) Let \(p=5\) and \(m=3.\) Then \(q=125\) and \(n=3076.\) By Theorem 2, the code \(\textit{C}_{D}\) is a [3076, 6, 2352] linear code. Its weight enumerator is
which is checked by Magma.
4 Concluding remarks
Let \(w_{min}\) and \(w_{max}\) be the minimum and maximum nonzero weight of linear code \(\textit{C}_{D}\), respectively. We recall that if
then all nonzero codewords of code \(\textit{C}_{D}\) are minimal (see [8]).
By Theorem 1, we easily check
where even \(m\ge 4\). Moreover, by Theorem 2 we easily check
where odd \(m\ge 3\).
Hence, the linear codes in this paper satisfy \(w_{min}/w_{max}> (p-1)/p\) for \(m\ge 3\), and can be used to get secret sharing schemes with interesting access structures.
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The authors are very grateful to the editor and the anonymous reviewers for their valuable comments and suggestions to improve the quality of this paper.
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J. Ahn was financially supported by a research fund of Chungnam National University in 2015.
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Ahn, J., Ka, D. Weight enumerators of a class of linear codes. AAECC 29, 59–76 (2018). https://doi.org/10.1007/s00200-017-0329-8
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DOI: https://doi.org/10.1007/s00200-017-0329-8