Linearity and classification of ZpZp2-linear generalized Hadamard codes

The ${\mathbb{Z}}_p{\mathbb{Z}}_{p^2}$-additive codes are subgroups of ${\mathbb{Z}}_p^{{\alpha }_1}\times {\mathbb{Z}}_{p^2}^{{\alpha }_2}$, and can be seen as linear codes over ${\mathbb{Z}}_p$ when ${\alpha }_2=0$, ${\mathbb{Z}}_{p^2}$-additive codes when ${\alpha }_1=0$, or ${\mathbb{Z}}_2{\mathbb{Z}}_4$-additive codes when $p=2$. A ${\mathbb{Z}}_p{\mathbb{Z}}_{p^2}$-linear generalized Hadamard (GH) code is a GH code over ${\mathbb{Z}}_p$ which is the Gray map image of a ${\mathbb{Z}}_p{\mathbb{Z}}_{p^2}$-additive code. Recursive constructions of ${\mathbb{Z}}_p{\mathbb{Z}}_{p^2}$-additive GH codes of type $({\alpha }_1,{\alpha }_2;t_1,t_2)$ with $t_1,t_2\ge 1$ are known. In this paper, we generalize some known results for ${\mathbb{Z}}_p{\mathbb{Z}}_{p^2}$-linear GH codes with $p=2$ to any $p\ge 3$ prime when ${\alpha }_1\ne 0$, and then we compare them with the ones obtained when ${\alpha }_1=0$. First, we show for which types the corresponding ${\mathbb{Z}}_p{\mathbb{Z}}_{p^2}$-linear GH codes are nonlinear over ${\mathbb{Z}}_p$. Then, for these codes, we compute the kernel and its dimension, which allow us to classify them completely. Moreover, by computing the rank of some of these codes, we show that, unlike ${\mathbb{Z}}_4$-linear Hadamard codes, the ${\mathbb{Z}}_{p^2}$-linear GH codes are not included in the family of ${\mathbb{Z}}_p{\mathbb{Z}}_{p^2}$-linear GH codes with ${\alpha }_1\ne 0$ when $p\ge 3$ prime. Indeed, there are some families with infinite nonlinear ${\mathbb{Z}}_p{\mathbb{Z}}_{p^2}$-linear GH codes, where the codes are not equivalent to any ${\mathbb{Z}}_{p^s}$-linear GH code with $s\ge 2$.     

Asymptotically good ZprZps-additive cyclic codes

We construct a class of ${\mathbb{Z}}_{p^r}{\mathbb{Z}}_{p^s}$-additive cyclic codes generated by pairs of polynomials, where p is a prime number. Based on probabilistic arguments, we determine the asymptotic rates and relative distances of this class of codes: the asymptotic Gilbert-Varshamov bound at $\frac{1+p^{s-r}}{2}\delta$ is greater than $\frac{1}{2}$ and the relative distance of the code is convergent to δ, while the rate is convergent to $\frac{1}{1+p^{s-r}}$ for $0<\delta <\frac{1}{1+p^{s-r}}$ and $1\le r<s$. As a consequence, we prove that there exist numerous asymptotically good ${\mathbb{Z}}_{p^r}{\mathbb{Z}}_{p^s}$-additive cyclic codes.     

On the symbol-pair distances of repeated-root constacyclic codes of length 2ps

Let $p$ be an odd prime, and $\lambda$ be a nonzero element of the finite field ${\mathbb{F}}_{p^m}$. The $\lambda$-constacyclic codes of length $2p^s$ over ${\mathbb{F}}_{p^m}$ are classified as the ideals of quotient ring $\frac{{\mathbb{F}}_{p^m}\left[x\right]}{〈x^{2p^s}?\lambda 〉}$ in terms of their generator polynomials. Based on these generator polynomials, the symbol-pair distances of all such $\lambda$-constacyclic codes of length $2p^s$ are obtained in this paper. As an application, all MDS symbol-pair constacyclic codes of length $2p^s$ over ${\mathbb{F}}_{p^m}$ are established, which produce many new MDS symbol-pair codes with good parameters.     

Construction of extremal Type II Z2k-codes

We give methods for constructing many self-dual ${\mathbb{Z}}_m$-codes and Type II ${\mathbb{Z}}_{2k}$-codes of length 2n starting from a given self-dual ${\mathbb{Z}}_m$-code and Type II ${\mathbb{Z}}_{2k}$-code of length 2n, respectively. As an application, we construct extremal Type II ${\mathbb{Z}}_{2k}$-codes of length 24 for $k=4,5,\dots ,20$ and extremal Type II ${\mathbb{Z}}_{2k}$-codes of length 32 for $k=4,5,\dots ,10$. We also construct new extremal Type II ${\mathbb{Z}}_4$-codes of lengths 56 and 64.     

Permutation binomials of the form xr(xq−1 + a) over Fqe

We present several existence and nonexistence results for permutation binomials of the form $x^r(x^{q-1}+a)$, where $e\ge 2$ and $a\in {\mathbb{F}}_{q^e}^{⁎}$. As a consequence, we obtain a complete characterization of such permutation binomials over ${\mathbb{F}}_{q^2}$, ${\mathbb{F}}_{q^3}$, ${\mathbb{F}}_{q^4}$, ${\mathbb{F}}_{p^5}$, and ${\mathbb{F}}_{p^6}$, where p is an odd prime.     

Time-frequency analysis on the adeles over the rationals

We show that the construction of Gabor frames in $L^2(\mathbb{R})$ with generators in ${\mathbf{S}}_0(\mathbb{R})$ and with respect to time-frequency shifts from a rectangular lattice $\alpha \mathbb{Z}\times \beta \mathbb{Z}$ is equivalent to the construction of certain Gabor frames for $L^2$ over the adeles over the rationals and the group $\mathbb{R}\times {\mathbb{Q}}_p$. Furthermore, we detail the connection between the construction of Gabor frames on the adeles and on $\mathbb{R}\times {\mathbb{Q}}_p$ with the construction of certain Heisenberg modules.     

Galois LCD codes over mixed alphabets

In this work we give a characterization of Galois Linear Complementary Dual codes and Galois-invariant codes over mixed alphabets of finite chain rings, which leads to the study of the Gray image of ${\mathbb{F}}_p{\mathbb{F}}_p[\theta ]$-linear codes, where $p\in \{2;3\}$ and $\theta \ne {\theta }^2=0$ that provides LCD codes over ${\mathbb{F}}_p$.     

The dual code of any (δ+αu2)-constacyclic code over F2m[u]∕〈u4〉 of oddly even length

Let ${\mathbb{F}}_{2^m}$ be a finite field of cardinality $2^m$, $R={\mathbb{F}}_{2^m}\left[u\right]∕〈u^4〉={\mathbb{F}}_{2^m}+u{\mathbb{F}}_{2^m}+u^2{\mathbb{F}}_{2^m}+u^3{\mathbb{F}}_{2^m}$ $(u^4=0)$ which is a finite chain ring, and $n$ is an odd positive integer. For any $\delta ,\alpha \in {\mathbb{F}}_{2^m}^{\times }$, an explicit representation for the dual code of any $(\delta +\alpha u^2)$-constacyclic code over $R$ of length $2n$ is given. And some dual codes of $(1+u^2)$-constacyclic codes over $R$ of length 14 are constructed. For the case of $\delta =1$, all distinct self-dual $(1+\alpha u^2)$-constacyclic codes over $R$ of length $2n$ are determined.