Abstract
Let <inline-formula> <tex-math notation="LaTeX">$G$ </tex-math></inline-formula> be an additive group of order <inline-formula> <tex-math notation="LaTeX">$v$ </tex-math></inline-formula>. A <inline-formula> <tex-math notation="LaTeX">$k$ </tex-math></inline-formula>-element subset <inline-formula> <tex-math notation="LaTeX">$D$ </tex-math></inline-formula> of <inline-formula> <tex-math notation="LaTeX">$G$ </tex-math></inline-formula> is called a <inline-formula> <tex-math notation="LaTeX">$(v, k, \lambda, t)$ </tex-math></inline-formula>-almost difference set if the expressions <inline-formula> <tex-math notation="LaTeX">$g-h$ </tex-math></inline-formula>, for <inline-formula> <tex-math notation="LaTeX">$g$ </tex-math></inline-formula> and <inline-formula> <tex-math notation="LaTeX">$h$ </tex-math></inline-formula> in <inline-formula> <tex-math notation="LaTeX">$D$ </tex-math></inline-formula>, represent <inline-formula> <tex-math notation="LaTeX">$t$ </tex-math></inline-formula> of the non-identity elements in <inline-formula> <tex-math notation="LaTeX">$G$ </tex-math></inline-formula> exactly <inline-formula> <tex-math notation="LaTeX">$\lambda $ </tex-math></inline-formula> times and every other non-identity element <inline-formula> <tex-math notation="LaTeX">$\lambda + 1$ </tex-math></inline-formula> times. Almost difference sets are highly sought after as they can be used to produce functions with optimal nonlinearity, cyclic codes, and sequences with three-level autocorrelation. A set of positive integers <inline-formula> <tex-math notation="LaTeX">$A$ </tex-math></inline-formula> is called a Golomb ruler if the difference between two distinct elements of <inline-formula> <tex-math notation="LaTeX">$A$ </tex-math></inline-formula> are different. In this paper, we use Singer type Golomb rulers to construct new families of almost difference sets. Additionally, we constructed 2-adesigns from these almost difference sets.
Highlights
Difference sets are a well-known class of mathematical objects used in the construction of designs and other combinatorial structures
ALMOST DIFFERENCE SETS VIA SINGER TYPE GOLOMB RULERS we describe three new constructions of almost difference sets from Singer type Golomb rulers
CONSTRUCTION 1 The following theorem shows how to construct an almost difference set from a Singer type Golomb ruler using homomorphic projection
Summary
Difference sets are a well-known class of mathematical objects used in the construction of designs and other combinatorial structures. The second construction is obtained by adding a new element to the Golomb ruler and yields (q2 + q + 1, q + 2, 1, (q − 2)(q + 1))-ADSs in cyclic groups of order q2 + q + 1 for all prime power q. The third construction is obtained by removing an element of the Golomb ruler and yields (q2 + q + 1, q, 0, 2q)-ADSs in cyclic groups of order q2 + q + 1 for all prime power q.
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