arXiv subjects
Publications and source records attributed to Ivan Landjev.
In this paper, we prove an analogue of the Erd\H{o}s-Ko-Rado theorem intersecting families of subspaces in projective Hjelmslev geometries over finite chain rings of nilpotency index 2. We give an example of maximal families that are not canonically intersectng.
This text contains some notes on the Griesmer bound. In particular, we give a geometric proof of the Griesmer bound for the generalized weights and show that a Solomon--Stiffler type construction attains it if the minimum distance is sufficiently large. We also determine the parameters of optimal binary codes for dimensions at most seven and the optimal ternary codes for dimensions at most five.
Let $C$ be a binary code of length $n$ with distances $0<d_1<\cdots<d_s\le n$. In this note we prove a general upper bound on the size of $C$ without any restriction on the distances $d_i$. The bound is asymptotically optimal.
In this paper we prove a new reducibility result for mini-hypers in projective geometries over finite fields. It is further used to characterize the minihypers with parameters (70, 22) in PG(4, 3). The latter can be used to attack the existence problem for some hypothetical ternary Griesmer codes of dimension 6.
We present various new constructions and bounds for arcs in projective Hjelmslev planes over finite chain rings of nilpotency index 2. For the chain rings of cardinality at most 25 we give updated tables with the best known upper and lower bounds for the maximum size of such arcs.
We consider the problem of finding $A_2(n,\{d_1,d_2\})$ defined as the maximal size of a binary (non-linear) code of length $n$ with two distances $d_1$ and $d_2$. Binary codes with distances $d$ and $d+2$ of size $\sim\frac{n^2}{\frac{d}{2}(\frac{d}{2}+1)}$ can be obtained from $2$-packings of an $n$-element set by blocks of cardinality $\frac{d}{2}+1$. This value is far from the upper bound $A_2(n,\{d_1,d_2\})\le1+{n\choose2}$ proved recently by Barg et al. In this paper we prove that for every fixed $d$ ($d$ even) there exists an integer $N(d)$ such that for every $n\ge N(d)$ it holds $A_2(n,\{d,d+2\})=D(n,\frac{d}{2}+1,2)$, or, in other words, optimal codes are isomorphic to constant weight codes. We prove also estimates on $N(d)$ for $d=4$ and $d=6$.
In this paper, we study the algebraic structure of $(\sigma,\delta)$-polycyclic codes, defined as submodules in the quotient module $S/Sf$, where $S=R[x,\sigma,\delta]$ is the Ore extension ring, $f\in S$, and $R$ is a finite but not necessarily commutative ring. We establish that the Euclidean duals of $(\sigma,\delta)$-polycyclic codes are $(\sigma,\delta)$-sequential codes. By using $(\sigma,\delta)$-Pseudo Linear Transformation, we define the annihilator dual of $(\sigma,\delta)$-polycyclic codes. Then, we demonstrate that the annihilator duals of $(\sigma,\delta)$-polycyclic codes maintain their $(\sigma,\delta)$-polycyclic nature. Furthermore, we classify when two $(\sigma,\delta)$-polycyclic codes are Hamming isometrical equivalent. By employing Wedderburn polynomials, we introduce simple-root $(\sigma,\delta)$-polycyclic codes. Subsequently, we define the $(\sigma, \delta)$-Mattson-Solomon transform for this class of codes and we address the problem of decomposing these codes by using the properties of Wedderburn polynomials.
In this paper, we give a geometric construction of the three strong non-lifted $(3\mod{5})$-arcs in $\operatorname{PG}(3,5)$ of respective sizes 128, 143, and 168, and construct an infinite family of non-lifted, strong $(t\mod{q})$-arcs in $\operatorname{PG}(r,q)$ with $t=(q+1)/2$ for all $r\ge3$ and all odd prime powers $q$.
The proof of the non-existence of Griesmer $[104, 4, 82]_5$-codes is just one of many examples where extendability results are used. In a series of papers Landjev and Rousseva have introduced the concept of $(t\operatorname{mod} q)$-arcs as a general framework for extendability results for codes and arcs. Here we complete the known partial classification of $(3 \operatorname{mod} 5)$-arcs in $\operatorname{PG}(3,5)$ and uncover two missing, rather exceptional, examples disproving a conjecture of Landjev and Rousseva. As also the original non-existence proof of Griesmer $[104, 4, 82]_5$-codes is affected, we present an extended proof to fill this gap.