Nikita Polyanskii

IOTA Foundation

In this work, we consider $q$-ary signature codes of length $k$ and size $n$ for a noisy adder multiple access channel. A signature code in this model has the property that any subset of codewords can be uniquely reconstructed based on any vector that is obtained from the sum (over integers) of these codewords. We show that there exists an algorith...

Lifted Reed-Solomon and multiplicity codes are classes of codes, constructed from specific sets of $m$-variate polynomials. These codes allow for the design of high-rate codes that can recover every codeword or information symbol from many disjoint sets. Recently, the underlying approaches have been combined for the bi-variate case to construct lif...

In this work, we introduce a natural notion concerning finite vector spaces. A family of k-dimensional subspaces of Fqn, which forms a partial spread, is called almost affinely disjoint if any (k+1)-dimensional subspace containing a subspace from the family non-trivially intersects with only a few subspaces from the family. The central question dis...

We present a list decoding algorithm for \(\mathbb{F}_q\)-linear codes that generalize the Reed–Solomon \(s\)-codes.

Lifted Reed-Solomon and multiplicity codes are classes of codes, constructed from specific sets of m-variate polynomials. These codes allow for the design of high-rate codes that can recover every codeword or information symbol from many disjoint sets. Recently, the underlying approaches have been combined for the bi-variate case to construct lifte...

Lifted codes are a class of evaluation codes attracting more attention due to good locality and intermediate availability. In this work we introduce and study quadratic-curve-lifted Reed-Solomon (QC-LRS) codes, where the codeword symbols whose coordinates are on a quadratic curve form a codeword of a Reed-Solomon code. We first develop a necessary...

We consider the problem of constructing codes that can correct deletions that are localized within a certain part of the codeword that is unknown a priori. Namely, the model that we study is when at most $k$ deletions occur in a window of size $k$, where the positions of the deletions within this window are not necessarily consecutive. Localized de...

This paper is a collection of results on combinatorial properties of codes for the Z-channel. A Z-channel with error fraction $\tau$ takes as input a length-$n$ binary codeword and injects in an adversarial manner $n\tau$ asymmetric errors, i.e., errors that only zero out bits but do not flip $0$'s to $1$'s. It is known that the largest $(L-1)$-lis...

We prove that the maximum number of words in a code that corrects a fraction of \(1/4+\varepsilon\) of asymmetric errors in a Z-channel is \(\Theta(\varepsilon^{-3/2})\) as \(\varepsilon\to 0\).

We consider the problem of coding for the substring channel, in which information strings are observed only through their (multisets of) substrings. Because of applications to DNA-based data storage, due to DNA sequencing techniques, interest in this channel has renewed in recent years. In contrast to existing literature, we consider a noisy channe...

In this paper, we discuss two-stage encoding algorithms capable of correcting a fraction of asymmetric errors. Suppose that we can transmit $n$ binary symbols $(x_1,\ldots,x_n)$ one-by-one over the Z-channel, in which a 1 is received if and only if a 1 is transmitted. At some moment, say $n_1$, it is allowed to use the complete feedback of the chan...

In this paper, a new problem of transmitting information over the adversarial insertion-deletion channel with feedback is introduced. Suppose that we can transmit $n$ binary symbols one-by-one over the channel, in which some symbols can be deleted and some additional symbols can be inserted. After each transmission, the encoder is notified about th...

Lifted Reed-Solomon codes, a subclass of lifted affine-invariant codes, have been shown to be of high rate while preserving locality properties similar to generalized Reed-Muller codes, which they contain as subcodes. This work introduces a simple bounded distance decoder for (subcodes of) lifted affine-invariant codes that is guaranteed to decode...

In this paper, we derive the exact weight distributions that emerge during each stage of successive cancellation decoding of polar codes. Though we do not compute the distance spectrum of polar codes, the results allow us to get an estimate of the decoding error probability and to show a link between the first nonzero components of the weight distr...

In this paper, we consider encoding strategies for the Z-channel with noiseless feedback. We analyze the asymptotic case where the maximal number of errors is proportional to the blocklength, which goes to infinity. Without feedback, the asymptotic rate of error-correcting codes for the error fraction \(\tau \ge 1/4\) is known to be zero. It was al...

Lifted Reed-Solomon codes and multiplicity codes are two classes of evaluation codes that allow for the design of high-rate codes that can recover every codeword or information symbol from many disjoint sets. Recently, the underlying approaches have been combined to construct lifted bi-variate multiplicity codes, that can further improve on the rat...

A primitive k-batch code encodes a string x of length n into a string y of length N, such that each multiset of k symbols from x has k mutually disjoint recovering sets from y. In this paper, we discuss new constructions of binary primitive batch codes. First, we develop novel explicit and random coding constructions of linear primitive batch codes...

In this paper, we consider encoding strategies for the Z-channel with noiseless feedback. We analyze the asymptotic case where the maximal number of errors is proportional to the blocklength, which goes to infinity. Without feedback, the asymptotic rate of error-correcting codes for the error fraction $\tau\ge 1/4$ is known to be zero. It was also...

In this work, we introduce a natural notion concerning vector finite spaces. A family of $k$-dimensional subspaces of $\mathbb{F}_q^n$ is called almost affinely disjoint if any $(k+1)$-dimensional subspace containing a subspace from the family non-trivially intersects with only a few subspaces from the family. The central question discussed in the...

Guo, Kopparty and Sudan have initiated the study of error-correcting codes derived by lifting of affine-invariant codes. Lifted Reed-Solomon (RS) codes are defined as the evaluation of polynomials in a vector space over a field by requiring their restriction to every line in the space to be a codeword of the RS code. In this paper, we investigate l...

In this paper, we present an efficiently encodable and decodable code construction that is capable of correction a burst of deletions of length at most $k$. The redundancy of this code is $\log n + k(k+1)/2\log \log n+c_k$ for some constant $c_k$ that only depends on $k$ and thus is scaling-optimal. The code can be split into two main components. F...

We study the duplication with transposition distance between strings of length $n$ over a $q$-ary alphabet and their roots. In other words, we investigate the number of duplication operations of the form $x = (abcd) \to y = (abcbd)$, where $x$ and $y$ are strings and $a$, $b$, $c$ and $d$ are their substrings, needed to get a $q$-ary string of leng...

The conventional model of disjunctive group testing assumes that there are several defective elements (or defectives) among a large population, and a group test yields the positive response if and only if the testing group contains at least one defective element. The basic problem is to find all defectives using a minimal possible number of group t...

In this paper, we derive the exact weight distributions for the successive cancellation decoding of polar codes. The results allow to get an estimate of the decoding error probability and to show a link between the first nonzero components of the weight distribution and the partial order between the synthetic channels. Furthermore, we prove a state...

A binary matrix is called an s-separable code for the disjunctive multiple-access channel (disj-MAC) if Boolean sums of sets of s columns are all distinct. The well-known issue of the combinatorial coding theory is to obtain upper and lower bounds on the rate of s-separable codes for the disj-MAC. In our paper, we generalize the problem and discuss...

A primitive $k$-batch code encodes a string $x$ of length $n$ into string $y$ of length $N$, such that each multiset of $k$ symbols from $x$ has $k$ mutually disjoint recovering sets from $y$. We develop new explicit and random coding constructions of linear primitive batch codes based on finite geometry. In some parameter regimes, our proposed cod...

Provided is a system and method for determining a generalized LDPC code for forward error correction channel coding that has a repeat-accumulate code structure to allow for easy encoding. Cordaro-Wagner component code check matrices may be selected, wherein each of the selected Cordora-Wagner component code check matrices has two rows which replace...

Provided is an encoder, a decoder, a computer-readable medium and methods of forward error correction channel encoding/decoding within a HARQ scheme, based on a generalized quasi-cyclic low-density parity-check code comprising a Cordaro-Wagner component code.

The exact values of the optimal symmetric rate point in the Cover–Leung capacity region of the two-user union channel with complete feedback were determined by Willems when the size of the input alphabet is 2, and by Vinck, Hoeks and Post when the size is at least 6. We complete this line of research when the size of the input alphabet is 3, 4 or 5...

The given paper presents a method for constructing a QC-LDPC code of shorter length by length adaption from a given QC-LDPC code of maximal length. The proposed method can be considered as a generalization of floor lifting. Making some offline calculation it is possible to construct a sequence of QC-LDPC codes with different circulant sizes generat...

The given paper presents several probabilistic statements for the methods related to constructing a QC-LDPC code of shorter length by length adaption from a given QC-LDPC code of maximal length. Several probabilistic statements concerning a theoretical improvement of the generalization of floor lifting method with respect to the number of small cyc...

The exact values of the optimal symmetric rate point in the Cover--Leung capacity region of the two-user union channel with complete feedback were determined by Willems when the size of the input alphabet is 2, and by Vinck, Hoeks and Post when the size is at least 6. We complete this line of research when the size of the input alphabet is 3, 4 or...

A set of vertices $S$ resolves a graph if every vertex is uniquely determined by its vector of distances to the vertices in $S$. The metric dimension of a graph is the minimum cardinality of a resolving set of the graph. Fix a connected graph $G$ on $q \ge 2$ vertices, and let $M$ be the distance matrix of $G$. We prove that if there exists $w \in...

In the given paper we present a novel approach for constructing a QC-LDPC code of smaller length by lifting a given QC-LDPC code. The proposed method can be considered as a generalization of floor lifting. Also we prove several probabilistic statements concerning a theoretical improvement of the method with respect to the number of small cycles. Ma...

The conventional model of disjunctive group testing assumes that there are several defective elements (or defectives) among t items, and a group test yields the positive response if and only if the testing group contains at least one defective element. The basic problem is to find all defectives using a minimal possible number of group tests. Howev...

A binary matrix is called an s-separable code for the disjunctive multiple-access channel (disj-MAC) if Boolean sums of sets of s columns are all distinct. The well-known issue of the combinatorial coding theory is to obtain upper and lower bounds on the rate of s-separable codes for disj-MAC. In our paper, we generalize the problem and discuss upp...

A binary code is said to be a disjunctive list-decoding $s_L$-code, $s\ge1$, $L\ge1$, if the code is identified by the incidence matrix of a family of finite sets in which the union (or disjunctive sum) of any $s$ sets can cover not more than $L-1$ other sets of the family. In this paper, we consider a similar class of binary codes which are %symme...

An $s$-subset of columns of a binary code is said to be an $(s,\ell)$-bad subset of columns if there exists a subset of other $\ell$ columns in the code such that the conjunction of $\ell$ columns is covered by the disjunctive sum of $s$ columns. A binary code is called a cover-free $(s,\ell)$-code if there is no $(s,\ell)$-bad subset of columns in...

We discover some important properties of cover-free (CF) codes, separating system (SS) codes and completely separating system (CSS) codes connected with the concept of constant weight CF codes. New upper and lower bounds on the rate of CF codes, SS codes and CSS codes based on the known results for CF codes are obtained. Tables of numerical values...

The invention relates to a device (100) and method (1100) for generating on the basis of a first protograph matrix P1 of size m x n, wherein the first protograph matrix P1 defines a first code H1 a second protograph matrix P2 of size (m + d) x (n + d), wherein the second protograph matrix P2 defines a second code H2. The device (100) comprises a pr...

A method for quasi-cyclic low-density parity-check (QC-LDPC) encoding and decoding of a data packet by a lifted matrix is provided, the method comprising: lifting the QC-LDPC code for maximal code length Nmax and maximal circulant size Z upper of the base matrix; generating a plurality of optimal values r, for a plurality of circulants Z 1, Z 2,......

Learning a hidden hypergraph is a natural generalization of the classical group testing problem that consists in detecting unknown hypergraph $H_{un}=H(V,E)$ by carrying out edge-detecting tests. In the given paper we focus our attention only on a specific family $\mathcal{F}(t,s,\ell)$ of localized hypergraphs for which the total number of vertice...

Let $1 \le s < t$, $N \ge 1$ be integers and a complex electronic circuit of size $t$ is said to be an $s$-active, $\; s \ll t$, and can work as a system block if not more than $s$ elements of the circuit are defective. Otherwise, the circuit is said to be an $s$-defective and should be replaced by a similar $s$-active circuit. Suppose that there e...

Group testing is a well known search problem that consists in detecting up to $s$ defective elements of the set $[t]=\{1,\ldots,t\}$ by carrying out tests on properly chosen subsets of $[t]$. In classical group testing the goal is to find all defective elements by using the minimal possible number of tests. In this paper we consider multistage grou...

We give a method of constructing a cover-free $(s, \ell)$-code. For $k > s$, our construction yields a $ {{n \choose s} \choose \ell}\times {n \choose k}$ cover-free $(s, \ell)$-code with a constant column weight.

A binary code is called a superimposed cover-free $(s,\ell)$-code if the code is identified by the incidence matrix of a family of finite sets in which no intersection of $\ell$ sets is covered by the union of $s$ others. A binary code is called a superimposed list-decoding $s_L$-code if the code is identified by the incidence matrix of a family of...

We say that an s-subset of codewords of a code X is (s, l)-bad if X contains l other codewords such that the conjunction of these l words is covered by the disjunction of the words of the s-subset. Otherwise, an s-subset of codewords of X is said to be (s, l)-bad. A binary code X is called a disjunctive (s, l) cover-free (CF) code if X does not con...

Let $1 \le s < t$, $N \ge 1$ be integers and a complex electronic circuit of size $t$ is said to be an $s$-active, $\; s \ll t$, and can work as a system block if not more than $s$ elements of the circuit are defective. Otherwise, the circuit is said to be an $s$-defective and should be substituted for the similar $s$-active circuit. Suppose that t...

Learning a hidden hypergraph is a natural generalization of the classical group testing problem that consists in detecting unknown hypergraph $H=(V,E)$ by carrying out edge-detecting tests. In the given paper we study a relaxation of this problem. We wish to identify \textit{almost} all hypergraphs by using the minimal number of tests. We focus our...

Group testing is a well known search problem that consists in detecting up to $s$ defective elements of the set $[t]=\{1,\ldots,t\}$ by carrying out tests on properly chosen subsets of $[t]$. In classical group testing the goal is to find all defective elements by using the minimal possible number of tests. In this paper we consider multistage grou...

DNA sequences are sequences with elements from the quaternary DNA alphabet {A, C, G, T}. An important property of them is their directedness and ability to form duplexes as a result of hybridization process, i.e., coalescing two oppositely directed sequences. In biological experiments exploiting this property it is necessary to generate an ensemble...

A binary code is said to be a disjunctive list-decoding $s_L$-code, $s\ge1$, $L\ge1$, (briefly, LD $s_L$-code) if the code is identified by the incidence matrix of a family of finite sets in which the union of any $s$ sets can cover not more than $L-1$ other sets of the family. In this paper, we introduce a natural {\em probabilistic} generalizatio...

A binary code is called a superimposed cover-free (s, ℓ)-code if the code is identified by the incidence matrix of a family of finite sets in which no intersection of ℓ sets is covered by the union of s others. A binary code is called a superimposed list-decoding sL-code if the code is identified by the incidence matrix of a family of finite sets i...

A binary code is said to be a disjunctive (s, ℓ) cover-free code if it is an incidence matrix of a family of sets where the intersection of any ℓ sets is not covered by the union of any other s sets of this family. A binary code is said to be a list-decoding disjunctive of strength s with list size L if it is an incidence matrix of a family of sets...

The concept of a generalized stem similarity function and the corresponding DNA codes are introduced. We give parameters for some optimal constructions called maximum distance separable DNA codes and obtain bounds on the maximum size of DNA codes.