A subspace of $\mathbb{F}_q^n$ is called cyclically covering if the union of $\sigma^i(U)$ can cover the whole space $\mathbb{F}_q^n$, where $\sigma$ is the cyclic shift, $0 \leqslant i \leqslant n-1$. Let $h_q(n)$ be the largest possible co-dimension of a cyclically covering subspace of $\mathbb{F}_q^n$. We show that $h_2(2p) = 2$ for every prime $p$ such that $2$ is a primitive root modulo $p$. By constacyclic codes, we show that $h_q((q-1)n) = 0$ when $h_q(n) = 0$ and $\gcd(n,q-1) = 1$. We also derive a lower bound on $h_q(n)$ by the concept of support weight distribution, which is important in coding theory. Finally, using irreducible cyclic codes, we present several families of $n$ such that $h_q(n) = 0$.
Let $n \geq 2$ be a positive integer, and $q$ be a prime power. We study the $number$ of additive decompositions of nonscalar matrices in the matrix ring $M_n(\mathbb{F}_q)$ as sums of elements from two prescribed conjugacy classes. Let $z \in M_n(\mathbb{F}_q)$ be nonscalar. We show that, except for the case $(n,q,\te...
Let $p$ be a prime and $\mathbb{F}_q/\mathbb{F}_r$ a finite field extension with $q=p^m$ and $r=p^s$. For any $k\mid q-1$, we consider $r$-ary irreducible cyclic codes (ICC) of the form $C(k,q/r) = \{(Tr_{q/r}(\gamma \omega^{ik})_{i=0}^{n-1})\}_{\gamma \in \mathbb{F}_r}$, with $\omega$ a primitive element of $\mathbb{F...
Ricardo A. Podestá, Denis E. Videla· 1 citation· ⚡1
Given a cyclic code $C_g = \langle g(x) \rangle$ of odd length $m$ over $\mathbb{Z}_4$, one common way to build a quasi-cyclic (QC) code is to pick $f_1, \dots, f_\ell \in \mathbb{Z}_4[x]$ and let $C = \langle (f_1 g, \dots, f_\ell g) \rangle$. Because $\mathbb{Z}_4$ is not a field, the type of $C_g$ ($4^{k_1}2^{k_2}$)...
Let $c\phi_k(n)$ denote the number of $k$-colored generalized Frobenius partitions of $n$. We prove that, for every $m\geq2$ and every $k\equiv2\pmod{2^m}$, \[ \sum_{n\geq0}c\phi_k(n)q^n\equiv\frac{\varphi(q)\,(q^2;q^2)_\infty}{(q;q)_\infty^2}\sum_{n\geq0}c\phi_{k/2}(n)q^{2n}\pmod{2^m}, \] where $\varphi(q)$ is the cla...
Let $E_n$ denote the number of alternating permutations of $\{1,\dots,n\}$, equivalently characterized by $\sum_{n\ge0}E_nz^n/n!=\sec z+\tan z$. For every $q\ge1$, the sequence $(E_n\bmod q)_{n\ge0}$ is eventually periodic; let $d(q)$ and $s(q)$ denote its minimal eventual period and preperiod. For every odd prime $p$,...
Let $R_2 = \mathbb{F}_{p^m} + u\mathbb{F}_{p^m}$ ($u^2 = 0$), where $p$ is an odd prime and $m, s \in \mathbb{N}$. For $p \equiv 3, 5 \pmod 7$ with $\gcd(m, 6) = 1$, the cyclotomic polynomial $\Phi_7(x)$ is irreducible over $\mathbb{F}_{p^m}$. This yields a direct sum decomposition $C = C_1 \oplus C_2$ for any cyclic c...
Payel Chandra, K. Hansda· 0 citations
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