American Institute of Mathematical Sciences

Two bounds on the minimum distance of cyclic codes are proposed. The first one generalizes the Roos bound by embedding the given cyclic code into a cyclic product code. The second bound also uses a second cyclic code, namely the non-zero-locator code, but is not directly related to cyclic product codes and it generalizes a special case of the Roos bound.

Applied in consumer electronics, communication, data storage system, secret sharing, authentication codes, association schemes, and strongly regular graphs, linear codes attract much interest. We consider the construction of linear codes with two or three weights. Let \begin{document}$ m_1,\ldots, m_t $\end{document} be \begin{document}$ t $\end{document} positive integers and \begin{document}$ T = \mathbb{F}_{q_1}\times \cdots \times \mathbb{F}_{q_t} $\end{document}, where \begin{document}$ q_i = p^{m_i} $\end{document} for \begin{document}$ 1\leq i\leq t $\end{document} and \begin{document}$ p $\end{document} is an odd prime. A linear code

\begin{document}$ \begin{equation*} \mathcal{C}_D = \{ \mathbf{c}(\mathbf{a}): \mathbf{a} = (a_1,\ldots,a_t)\in T\}, \end{equation*} $\end{document}

can be constructed by a defining set \begin{document}$ D $\end{document}, where \begin{document}$ D $\end{document} is a subset of \begin{document}$ T $\end{document} and \begin{document}$ \mathbf{c}(\mathbf{a}) = (\sum_{i = 1}^{t}\mathrm{Tr}_1^{m_i}(a_ix_i))_{ \mathbf{x} = (x_1,\ldots,x_t)\in D} $\end{document}. We construct linear codes with two or three weights from the following three defining sets:

● \begin{document}$ D_0 = \{\mathbf{x}\in T\backslash \{\mathbf{0}\}: \sum_{i = 1}^t\mathrm{Tr}_1^{m_i}(x_i^2) = 0\} $\end{document},

● \begin{document}$ D_{SQ} = \{\mathbf{x}\in T: \sum_{i = 1}^t\mathrm{Tr}_1^{m_i}(x_i^2)\in SQ\} $\end{document},

● \begin{document}$ D_{NSQ} = \{\mathbf{x}\in T: \sum_{i = 1}^t\mathrm{Tr}_1^{m_i}(x_i^2)\in NSQ\} $\end{document},

where \begin{document}$ SQ $\end{document} is the set of all the squares in \begin{document}$ \mathbb{F}_p^* $\end{document} and \begin{document}$ NSQ $\end{document} is the set of all the nonsquares in \begin{document}$ \mathbb{F}_p^* $\end{document}. We also determine the weight distributions of these codes. The punctured codes of codes from the defining set \begin{document}$ D_0 $\end{document} contain optimal codes meeting certain bounds. This paper generalizes results of [22].

Golay complementary sets (GCSs) are widely used in different communication systems, i.e., GCSs could be used in OFDM systems to control peak-to-mean envelope power ratio (PMEPR). In this paper, inspired by the work on GCSs with large zero correlation zone given by Chen et al in 2018, we investigate the relationship between GCSs and zero odd-periodic correlation zone (ZOCZ) sequence sets, and present GCSs with flexible sequence set sizes, sequence lengths, large ZOCZ and low PMEPR. Those proposed sequences could be applied in OFDM system for synchronization.

We introduce the Hadamard full propelinear codes that factorize as direct product of groups such that their associated group is \begin{document}$ C_{2t}\times C_2 $\end{document}. We study the rank, the dimension of the kernel, and the structure of these codes. For several specific parameters we establish some links from circulant Hadamard matrices and the nonexistence of the codes we study. We prove that the dimension of the kernel of these codes is bounded by \begin{document}$ 3 $\end{document} if the code is nonlinear. We also get an equivalence between circulant complex Hadamard matrix and a type of Hadamard full propelinear code, and we find a new example of circulant complex Hadamard matrix of order \begin{document}$ 16 $\end{document}.

Bent functions have many important applications in cryptography and coding theory. This paper considers a class of \begin{document}$ p $\end{document}-ary functions with the Dillon exponent of the form

\begin{document}$ f(x) = \sum\limits_{i = 0}^{q-1}(Tr^n_1(a_1x^{(r i+s)(q-1)})+Tr^n_1(a_2x^{(r i+s)(q-1)+\frac{q^2-1}{2}}))+bx^{\frac{q^2-1}{2}}, $\end{document}

where \begin{document}$ n = 2m $\end{document}, \begin{document}$ q = p^m $\end{document}, \begin{document}$ p $\end{document} is an odd prime, \begin{document}$ a_1,a_2\in \mathbb{F}_{p^n} $\end{document}, and \begin{document}$ b\in \mathbb{F}_p $\end{document}. With the help of Kloosterman sums, we present an explicit characterization of these \begin{document}$ p $\end{document}-ary regular bent functions for the case \begin{document}$ gcd(s-r,\frac{q+1}{2}) = 1 $\end{document} and \begin{document}$ gcd(r,q+1) = 1 $\end{document} or \begin{document}$ 2 $\end{document}. Our results generalize results of Li et al. [IEEE Trans. Inf. Theory 59 (2013) 1818-1831].

We present a cryptanalysis of a signature scheme HIMQ-3 due to Kyung-Ah Shim et al [10], which is a submission to National Institute of Standards and Technology (NIST) standardization process of post-quantum cryptosystems in 2017. We will show that inherent to the signing process is a leakage of information of the private key. Using this information one can forge a signature.

Let \begin{document}$ {\mathbb F}_q $\end{document} be the finite field with \begin{document}$ q = p^m $\end{document} elements, where \begin{document}$ p $\end{document} is an odd prime and \begin{document}$ m $\end{document} is a positive integer. Let \begin{document}$ \operatorname{Tr}_m $\end{document} denote the trace function from \begin{document}$ {\mathbb F}_q $\end{document} onto \begin{document}$ {\mathbb F}_p $\end{document}, and the defining set \begin{document}$ D\subset {\mathbb F}_q^t $\end{document}, where \begin{document}$ t $\end{document} is a positive integer. In this paper, the set \begin{document}$ D = \{(x_1, x_2, \cdots, x_t)\in {\mathbb F}_q^t:\operatorname{Tr}_m(x_1^2+x_2^2+\cdots+x_t^2) = 0, \operatorname{Tr}_m(x_1+x_2+\cdots+x_t) = 1\} $\end{document}. Define the \begin{document}$ p $\end{document}-ary linear code \begin{document}$ {\mathcal C}_D $\end{document} by

\begin{document}$ \begin{eqnarray*} {\mathcal C}_D = \{\textbf{c}(a_1, a_2, \cdots, a_t): (a_1, a_2, \cdots, a_t)\in {\mathbb F}_q^t\}, \end{eqnarray*} $\end{document}

where

\begin{document}$ \textbf{c}(a_1, a_2, \cdots, a_t) = (\operatorname{Tr}_m(a_1x_1+a_1x_2\cdots+a_tx_t))_{(x_1, \cdots, x_t)\in D}. $\end{document}

We evaluate the complete weight enumerator of the linear codes \begin{document}$ {\mathcal C}_D $\end{document}, and present its weight distributions. Some examples are given to illustrate the results.

We investigate a class of linear codes by choosing a proper defining set and determine their complete weight enumerators and weight enumerators. These codes have at most three weights and some of them are almost optimal so that they are suitable for applications in secret sharing schemes. This is a supplement of the results raised by Wang et al. (2017) and Kong et al. (2019).

In this work, we propose a post-quantum UC-commitment scheme in the Global Random Oracle Model, where only one non-programmable random oracle is available. The security of our proposal is based on two well-established post-quantum hardness assumptions from coding theory: The Syndrome Decoding and the Goppa Distinguisher. We prove that our proposal is perfectly hiding and computationally binding.

Let \begin{document}$ l $\end{document} be a prime with \begin{document}$ l\equiv 3\pmod 4 $\end{document} and \begin{document}$ l\ne 3 $\end{document}, \begin{document}$ N = l^m $\end{document} for \begin{document}$ m $\end{document} a positive integer, \begin{document}$ f = \phi(N)/2 $\end{document} the multiplicative order of a prime \begin{document}$ p $\end{document} modulo \begin{document}$ N $\end{document}, and \begin{document}$ q = p^f $\end{document}, where \begin{document}$ \phi(\cdot) $\end{document} is the Euler-function. Let \begin{document}$ \alpha $\end{document} be a primitive element of a finite field \begin{document}$ \Bbb F_{q} $\end{document}, \begin{document}$ C_0^{(N,q)} = \langle \alpha^N\rangle $\end{document} a cyclic subgroup of the multiplicative group \begin{document}$ \Bbb F_q^* $\end{document}, and \begin{document}$ C_i^{(N,q)} = \alpha^i\langle \alpha^N\rangle $\end{document} the cosets, \begin{document}$ i = 0,\ldots, N-1 $\end{document}. In this paper, we use Gaussian sums to obtain the explicit values of \begin{document}$ \eta_i^{(N, q)} = \sum_{x \in C_i^{(N,q)}}\psi(x) $\end{document}, \begin{document}$ i = 0,1,\cdots, N-1 $\end{document}, where \begin{document}$ \psi $\end{document} is the canonical additive character of \begin{document}$ \Bbb F_{q} $\end{document}. Moreover, we also compute the explicit values of \begin{document}$ \eta_i^{(2N, q)} $\end{document}, \begin{document}$ i = 0,1,\cdots, 2N-1 $\end{document}, if \begin{document}$ q $\end{document} is a power of an odd prime \begin{document}$ p $\end{document}.

As an application, we investigate the weight distribution of a \begin{document}$ p $\end{document}-ary linear code:

\begin{document}$ \mathcal{C}_{D} = \{C = ( \operatorname{Tr}_{q/p}(c x_1), \operatorname{Tr}_{q/p}(cx_2),\ldots, \operatorname{Tr}_{q/p}(cx_n)):c\in \Bbb{F}_{q}\}, $\end{document}

where its defining set \begin{document}$ D $\end{document} is given by

\begin{document}$ D = \{x\in \Bbb{F}_{q}^{*}: \operatorname{Tr}_{q/p}(x^{\frac{q-1}{l^{m}}}) = 0\} $\end{document}

and \begin{document}$ \operatorname{Tr}_{q/p} $\end{document} denotes the trace function from \begin{document}$ \Bbb F_{q} $\end{document} to \begin{document}$ \Bbb F_p $\end{document}.

In this paper, we give some properties of the cycle decomposition of a nonlinear feedback shift register called WG-NLFSR which was presented by Mandal and Gong recently. First we give the parity of the state transition transformation of WG-NLFSR and then by the relation of the parity of a permutation and its number of cycles given in Theorem 2 in Section 1, we show that the number of cycles in the cycle decomposition of WG-NLFSR is even. Second we study the properties of the cycle decomposition of WG-NLFSR when the coefficients of the characteristic polynomial belong to the proper subfields of the finite field on which the WG-NLFSR is defined. Finally, we give some properties of the cycle decomposition of the filtering WG7-NLFSR.

Let \begin{document}$ \mathbb{F}_{q} $\end{document} be a finite field with character \begin{document}$ p $\end{document} and \begin{document}$ p\neq{3},l\neq{3} $\end{document} be different odd primes. In this paper, we study constacyclic codes of length \begin{document}$ 6lp^s $\end{document} over finite field \begin{document}$ \mathbb{F}_{q} $\end{document}. The generator polynomials of all constacyclic codes and their duals are obtained. Moreover, we give the characterization and enumeration of linear complementary dual (LCD) and self-dual constacyclic codes of length \begin{document}$ 6lp^s $\end{document} over \begin{document}$ \mathbb{F}_{q} $\end{document}.

A \begin{document}$ (k,k-t) $\end{document}-SCID (set of Subspaces with Constant Intersection Dimension) is a set of \begin{document}$ k $\end{document}-dimensional vector spaces that have pairwise intersections of dimension \begin{document}$ k-t $\end{document}. Let \begin{document}$ \mathcal{C} = \{\pi_1,\ldots,\pi_n\} $\end{document} be a \begin{document}$ (k,k-t) $\end{document}-SCID. Define \begin{document}$ S: = \langle \pi_1, \ldots, \pi_n \rangle $\end{document} and \begin{document}$ I: = \langle \pi_i \cap \pi_j \mid 1 \leq i < j \leq n \rangle $\end{document}. We establish several upper bounds for \begin{document}$ \dim S + \dim I $\end{document} in different situations. We give a spectrum result under certain conditions for \begin{document}$ n $\end{document}, giving examples of \begin{document}$ (k,k-t) $\end{document}-SCIDs reaching a large interval of values for \begin{document}$ \dim S + \dim I $\end{document}.