Related papers: Turyn-type sequences: Classification, Enumeration …
Base sequences BS(n+1,n) are quadruples of {1,-1}-sequences (A;B;C;D), with A and B of length n+1 and C and D of length n, such that the sum of their nonperiodic autocorrelation functions is a delta-function. The base sequence conjecture,…
Base sequences BS(m,n) are quadruples (A;B;C;D) of {+1,-1}-sequences, with A and B of length m and C and D of length n, such that the sum of their nonperiodic autocorrelation functions is a delta-function. Normal sequences NS(n) are base…
The normal sequences NS(n) and near-normal sequences NN(n) play an important role in the construction of orthogonal designs and Hadamard matrices. They can be identified with certain base sequences (A;B;C;D), where A and B have length n+1…
Base sequences BS(m,n) are quadruples (A;B;C;D) of {+1,-1}-sequences, A and B of length m and C and D of length n, the sum of whose non-periodic auto-correlation functions is zero. Base sequences and some special subclasses of BS(n+1,n)…
A Barker sequence is a binary sequence for which all nontrivial aperiodic autocorrelations are at most 1 in magnitude. An old conjecture due to Turyn asserts that there is no Barker sequence of length greater than 13. In 1961, Turyn and…
Aperiodic autocorrelation is an important indicator of performance of sequences used in communications, remote sensing, and scientific instrumentation. Knowing a sequence's autocorrelation function, which reports the autocorrelation at…
We obtain new non-existence results of perfect p-ary sequences with period n (called type $[p, n]$). The first case is a class with type [p\equiv5\pmod 8,p^aqn']. The second case contains five types [p\equiv3\pmod 4,p^aq^ln'] for certain…
The base sequences BS(n+1,n) are four sequences of $\pm1$ and lengths n+1,n+1,n,n with zero auto correlation. The base sequence conjecture states that BS(n+1,n) exists for all positive integers and has been verified for $n\le40$. We present…
Tur\'{a}n number $T(n,5,3)$ is the minimum size of a system of triples out of a base set $X$ of $n$ elements such that every quintuple in $X$ contains a triple from the system. The exact values of $T(n,5,3)$ are known for $n \leq 17$.…
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…
A Transposition graph $T_n$ is defined as a Cayley graph over the symmetric group $Sym_n$ generated by all transpositions. It is known that the spectrum of $T_n$ consists of integers, but it is not known exactly how these numbers are…
A Barker sequence is a binary sequence for which all nontrivial aperiodic autocorrelations are either 0, 1 or -1. The only known Barker sequences have length 2, 3, 4, 5, 7, 11 or 13. It is an old conjecture that no longer Barker sequences…
We introduce a canonical form for near-normal sequences NN(n), and using it we enumerate the equivalence classes of such sequences for even n up to 30. These sequences are needed for Yang multiplication in the construction of longer…
Let $X_n = \{1,2,\dots,n\}$ be a finite set $(n\geq 2)$ and $T_n$ the full transformation semigroup on $X_n$. For a positive integer $l\leq n-1$, we define $$T_n(l) = \{\alpha\in T_n \colon \forall x,y\in X_n,\, |x-y| = l \;\Rightarrow\;…
An inversion sequence of length $n$ is an integer sequence $e=e_{1}e_{2}\dots e_{n}$ such that $0\leq e_{i}<i$ for each $i$. Corteel--Martinez--Savage--Weselcouch and Mansour--Shattuck began the study of patterns in inversion sequences,…
The Ap\'ery numbers may be defined by a cubic three-term recurrence relation, that is, a three-term relation where the coefficients are polynomials in the index of degree $3$. In this work, we first provide a systematic review of Ap\'ery…
This short note reports a master theorem on tight asymptotic solutions to divide-and-conquer recurrences with more than one recursive term: for example, T(n) = 1/4 T(n/16) + 1/3 T(3n/5) + 4 T(n/100) + 10 T(n/300) + n^2.
A $d$-subsequence of a sequence $\varphi = x_1\dots x_n$ is a subsequence $x_i x_{i+d} x_{i+2d} \dots$, for any positive integer $d$ and any $i$, $1 \le i \le n$. A \textit{$k$-Thue sequence} is a sequence in which every $d$-subsequence,…
We introduce hypergeometric-type sequences. They are linear combinations of interlaced hypergeometric sequences (of arbitrary interlacements). We prove that they form a subring of the ring of holonomic sequences. An interesting family of…
For Tur\'an's (3, 4)-conjecture, in the case of n = 3k+1 vertices, (.5)6^{k-1} non-isomorphic complexes are constructed that attain the conjecture. In the case of n = 3k+2 vertices, 6^{k-1} non-isomorphic complexes are constructed that…