Related papers: On de Bruijn Arrays Codes, Part I: Nonlinear Codes
For a set of integers $I$, we define a $q$-ary $I$-cycle to be a assignment of the symbols 1 through $q$ to the integers modulo $q^n$ so that every word appears on some translate of $I$. This definition generalizes that of de Bruijn cycles,…
The de Bruijn graph $G_K$ of a set of strings $S$ is a key data structure in genome assembly that represents overlaps between all the $K$-length substrings of $S$. Construction and navigation of the graph is a space and time bottleneck in…
Let $\widetilde{\alpha}$ be a length-$L$ cyclic sequence of characters from a size-$K$ alphabet $\mathcal{A}$ such that the number of occurrences of any length-$m$ string on $\mathcal{A}$ as a substring of $\widetilde{\alpha}$ is $\lfloor L…
Orientable sequences of order n are infinite periodic sequences with symbols drawn from a finite alphabet of size k with the property that any particular subsequence of length n occurs at most once in a period in either direction. They were…
The graph of overlapping permutations is a directed graph that is an analogue to the De Bruijn graph. It consists of vertices that are permutations of length $n$ and edges that are permutations of length $n+1$ in which an edge $a_1\cdots…
We give a complete characterization of simple graphs whose adjacency matrices generate binary linear complementary dual (LCD) codes. In particular, we completely characterize a distance-regular graph which yields an LCD code in terms of the…
A new family of binary linear completely transitive (and, therefore, completely regular) codes is constructed. The covering radius of these codes is growing with the length of the code. In particular, for any integer r > 1, there exist two…
In general, array codes consist of $m\times n$ arrays and in many cases, the arrays satisfy parity constraints along lines of different slopes (generally with a toroidal topology). Such codes are useful for RAID type of architectures, since…
A multifold $1$-perfect code ($1$-perfect code for list decoding) in any graph is a set $C$ of vertices such that every vertex of the graph is at distance not more than $1$ from exactly $\mu$ elements of $C$. In $q$-ary Hamming graphs,…
Given a parity-check matrix $H_m$ of a $q$-ary Hamming code, we consider a partition of the columns into two subsets. Then, we consider the two codes that have these submatrices as parity-check matrices. We say that anyone of these two…
Neural codes are collections of binary strings motivated by patterns of neural activity. In this paper, we study algorithmic and enumerative aspects of convex neural codes in dimension 1 (i.e. on a line or a circle). We use the theory of…
We show that any binary $(n=2^m-3, 2^{n-m}, 3)$ code $C_1$ is a part of an equitable partition (perfect coloring) $\{C_1,C_2,C_3,C_4\}$ of the $n$-cube with the parameters $((0,1,n-1,0)(1,0,n-1,0)(1,1,n-4,2)(0,0,n-1,1))$. Now the…
A binary linear code $C$ is a $\mathbb{Z}_2$-double cyclic code if the set of coordinates can be partitioned into two subsets such that any cyclic shift of the coordinates of both subsets leaves invariant the code. These codes can be…
A balanced generalized de Bruijn sequence with parameters $(n,l,k)$ is a cyclic sequence of $n$ bits such that (a) the number of 0's equals the number of 1's, and (b) each substring of length $l$ occurs at most $k$ times. We determine…
We investigate the class of completely regular codes in graphs with a distance partition C_0,..., C_\rho, where each set C_i, for 0<=i<=r-1, is an independent set. This work focuses on the existence problem for such codes in the…
Completely antipersistent binary time series are sequences in which every time that an $N$-bit string $\mu$ appears, the sequence is continued with a different bit than at the last occurrence of $\mu$. This dynamics is phrased in terms of a…
An $(n,R)$-covering sequence is a cyclic sequence whose consecutive $n$-tuples form a code of length $n$ and covering radius $R$. Using several construction methods improvements of the upper bounds on the length of such sequences for $n…
The nodes of the de Bruijn graph B(d,3) consist of all strings of length 3, taken from an alphabet of size d, with edges between words which are distinct substrings of a word of length 4. We give an inductive characterization of the maximum…
A complete classification of the perfect binary one-error-correcting codes of length 15 as well as their extensions of length 16 was recently carried out in [P. R. J. \"Osterg{\aa}rd and O. Pottonen, "The perfect binary one-error-correcting…
Nonlinear complexity is an important measure for assessing the randomness of sequences. In this paper we investigate how circular shifts affect the nonlinear complexities of finite-length binary sequences and then reveal a more explicit…