14 problems

Let $C$ be a polycyclic code with generator polynomial $g(x)$ satisfying … and suppose that $n$ is prime. Prime-length non-dual-containing conjecture. The code $C$ is not dual-cont…

Let $C$ be a polycyclic code over $\mathbb{F}_2$ generated by $g(x)$ dividing $x^n-ax^i-b$. Binary polycyclic-code isoduality conjecture. The code $C$ is iso-dual if and only if…

Let $v\geq 0$ and $\mathcal{T}\geq 1$, and consider binary polycyclic codes associated with … Here, an LCD code is a binary linear code whose intersection with its Euclidean dual i…

Binary isoduality characterization conjecture. The code $C$ is isodual if and only if

Let $t_1(x)=x^n-ax^i-b$ and $t_2(x)=x^n-a'x^{n-i}-b'$ satisfy … Let $S_1$ be the set of all polycyclic codes of length $n$ over $\mathbb{F}_q$ associated with $t_1(x)$, and let…

A polycyclic code over the binary field $\mathbb{F}_2$ may be self-dual, self-orthogonal, or dual-containing. Binary nonexistence conjecture. There is no self-dual, self-orthogonal…

Let $C$ be a polycyclic code over $\mathbb{F}_q$ with generator $g$. Self-dual polycyclic-code structure conjecture. The code $C$ is self-dual if and only if its generator has the…

Let $C$ be a polycyclic code generated by a polynomial $g(x)$ dividing $x^n-ax^i-b$, and suppose that $g^2(x)=x^n-ax^i-b$. The polycyclic-code isoduality conjecture. Then $C$ is is…

Let $C$ be a polycyclic code with generator polynomial $g$ dividing $x^n-ax^i-b$, where $n$ is prime. Prime-length dual-containing conjecture. The code $C$ is not dual-containing.…

Let $C$ be a polycyclic code over the binary field $\bbF_2$. Binary nonexistence conjecture. There are no polycyclic codes over $\bbF_2$ that are self-dual, self-orthogonal, or dua…

Let $C$ be a polycyclic code over the finite field $\bbF_q$, with generator polynomial $g$. Self-dual classification conjecture. The code $C$ is self-dual if and only if its genera…

Let $q$ be a prime power and consider the indicated trinomial-associated polycyclic codes. Iso-dual construction conjecture. The following codes are isodual: - If $q$ is a power of…

Let $C$ be a polycyclic code over a finite field, generated by a polynomial $g(x)$ dividing $x^n-ax^i-b$. Square-generator criterion. If … then $C$ is isodual. The surrounding text…

Let $t_1(x)=x^n-ax^i-b$ and $t_2(x)=x^n-a'x^{n-i}-b'$ be polynomials in $\bbF_q[x]$ with equal order, and let $C_1$ and $C_2$ denote the sets of length-$n$ polycyclic codes over…