In this article we consider the intersection graph G(R) of nontrivial proper ideals of a finite commutative principal ideal ring R with unity 1. Two distinct ideals are adjacent if they have non-trivial intersection. We characterize when the intersection graph is complete, bipartite, planar, Eulerian or Hamiltonian. We also find a formula to calculate the number of ideals in each ring and the d...

For any positive integers l and m, a set of integers is said to be (weakly) l-sumfree modulo m if it contains no (pairwise distinct) elements x1, x2, . . . , xl, y satisfying the congruence x1 + . . .+ xl ≡ y mod m. It is proved that, for any positive integers k and l, there exists a largest integer n for which the set of the first n positive integers {1, 2, . . . , n} admits a partition into k...

A general theory is presented of optimum diffusion via multi-dimensional 2point transforms over the ring of integers modulo a power of 2. This theory, which makes extensive use of graph-theoretic techniques, establishes the optimality of the transform diffuser used in the cipher SAFER+. A complete characterization is given of the matrix that describes an optimum transform diffuser. Finally, som...

Certain nonlinear binary single-error-correcting codes found by Julin, Best and others have simple descriptions as codes over the ring of integers modulo 4. ∗A different version of this paper appeared in: Designs, Codes and Cryptography, 41 (1994), 31–42. Quaternary Constructions for the Binary Single-Error-Correcting Codes of Julin, Best and Others J. H. Conway Mathematics Department Princeton...

A generic M-ary continuous phase modulation (CPM) scheme, with a modulation index of h = 1/M, can be decomposed into a continuous-phase encoder (CPE) followed by a memoryless modulator (MM), where the CPE is linear over the ring of integers modulo M. By using a channel encoder (CE), which is a convolutional encoder, to operate over the same ring of integers modulo M, the CE and the CPE can be c...

Let p be prime and X a Haar-random n × matrix over Z , the ring of -adic integers. P 1 ( t ) … l ∈ [ ] monic polynomials degree at most 2 whose images modulo are distinct irreducible in F where denotes finite field elements. For each j let G module / . We show that as goes to infinity, probabilities cok ≃ independent, probability can described terms Cohen–Lenstra distribution. also for any fixe...

A generic M-ary continuous phase modulation (CPM) scheme, with a modulation index of h = 1/M, can be modeled by a continuous-phase encoder (CPE) followed by a memoryless modulator (MM), where the CPE is linear over the ring of integers modulo M. By designing a channel encoder (CE), which is a convolutional encoder, to operate over the same ring of integers modulo M, the CE and the CPE can be co...