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

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Summary

INTRODUCTION

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.

ALMOST DIFFERENCE SETS VIA SINGER TYPE GOLOMB RULERS
CONSTRUCTION 3
CONCLUSION
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