A Family of Binary Sequences with Optimal Correlation Property and Large Linear Span

A family of binary sequences is presented and proved to have optimal correlation property and large linear span. It includes the small set of Kasami sequences, No sequence set and TN sequence set as special cases. An explicit lower bound expression on the linear span of sequences in the family is given. With suitable choices of parameters, it is proved that the family has exponentially larger linear spans than both No sequences and TN sequences. A class of ideal autocorrelation sequences is also constructed and proved to have large linear span.Comment: 21 page

Two-dimensional binary arrays with good autocorrelation

Calabro and Wolf (1968)Inform. Contr. 11) investigate the autocorrelation properties of certain periodic two-dimensional arrays. This note points out a relationship between periodic p × q-arrays with two-level autocorrelation and difference sets in the group C(p) × C(q), where C(n) denote the cyclic group of order n. This observation enables us to construct several families of such arrays, some of which are perfect

Linear Codes from Some 2-Designs

A classical method of constructing a linear code over \gf(q) with a $t$-design is to use the incidence matrix of the $t$-design as a generator matrix over \gf(q) of the code. This approach has been extensively investigated in the literature. In this paper, a different method of constructing linear codes using specific classes of $2$-designs is studied, and linear codes with a few weights are obtained from almost difference sets, difference sets, and a type of $2$-designs associated to semibent functions. Two families of the codes obtained in this paper are optimal. The linear codes presented in this paper have applications in secret sharing and authentication schemes, in addition to their applications in consumer electronics, communication and data storage systems. A coding-theory approach to the characterisation of highly nonlinear Boolean functions is presented

New constructions of signed difference sets

Signed difference sets have interesting applications in communications and coding theory. A $(v,k,\lambda)$-difference set in a finite group $G$ of order $v$ is a subset $D$ of $G$ with $k$ distinct elements such that the expressions $xy^{-1}$ for all distinct two elements $x,y\in D$, represent each non-identity element in $G$ exactly $\lambda$ times. A $(v,k,\lambda)$-signed difference set is a generalization of a $(v,k,\lambda)$-difference set $D$, which satisfies all properties of $D$, but has a sign for each element in $D$. We will show some new existence results for signed difference sets by using partial difference sets, product methods, and cyclotomic classes

Partial geometric designs and difference families

We examine the designs produced by different types of difference families. Difference families have long been known to produce designs with well behaved automorphism groups. These designs provide the elegant solutions desired for applications. In this work, we explore the following question: Does every (named) design have a difference family analogue? We answer this question in the affirmative for partial geometric designs

Realizations of Decimation Hadamard Transform for Special Classes of Binary Sequences with Two-level Autocorrelation

In an effort to search for a new binary two-level autocorrelation sequence, the decimation-Hadamard transform (DHT) based on special classes of known binary sequences with two-level autocorrelation is investigated. It is theoretically proved that some realizations of a binary generalized Gordon-Mills-Welch (GMW) sequence can be predicted from the structure of subfield factorization and the realization in its subfield. Furthermore, it is shown that the realization of any binary two-level autocorrelation sequence with respect to a quadratic residue (QR) sequence is either a QR sequence or the sequence itself. I