Zoning for Families

Is a group of eight unrelated adults and three children living together and sharing meals, household expenses, and responsibilities—and holding themselves out to the world to have long-term commitments to each other—a family? Not according to most zoning codes—including that of Hartford, Connecticut, where the preceding scenario presented itself a few years ago. Zoning, which is the local regulation of land use, almost always defines family, limiting those who may live in a dwelling unit to those who satisfy the zoning code’s definition. Often times, this definition is drafted in a way that excludes many modern living arrangements and preferences. This Article begins by exploring how zoning codes define both the family and the “functional family,” namely, a group of individuals living together like the Hartford group described above. The Article then carefully tracks judicial decisions that have rejected restrictive definitions of family and analyzes sociological and anthropological literature demonstrating that definitions excluding functional families are unreasonable as a matter of law. Based on the law as it has developed and demographic trends, my view is that governments must allow, but may regulate, functional families. The Article concludes with suggestions for local governments to revise their zoning codes to allow for functional families. In making these revisions, communities must weigh the real need to control density, the desire of functional families for privacy, and the urge to manage community character. Local governments who choose to regulate functional families may choose between three models of regulation: the density model, the privacy model, and the character model. Once decision-makers recognize these choices, they may more appropriately consider fellow community members’ increasingly diverse living arrangements and preferences—and better zone for families, whatever their modern form may entail

Hadamard Equiangular Tight Frames

An equiangular tight frame (ETF) is a type of optimal packing of lines in Euclidean space. They are often represented as the columns of a short, fat matrix. In certain applications we want this matrix to be flat, that is, have the property that all of its entries have modulus one. In particular, real flat ETFs are equivalent to self-complementary binary codes that achieve the Grey-Rankin bound. Some flat ETFs are (complex) Hadamard ETFs, meaning they arise by extracting rows from a (complex) Hadamard matrix. These include harmonic ETFs, which are obtained by extracting the rows of a character table that correspond to a difference set in the underlying finite abelian group. In this paper, we give some new results about flat ETFs. One of these results gives an explicit Naimark complement for all Steiner ETFs, which in turn implies that all Kirkman ETFs are possibly-complex Hadamard ETFs. This in particular produces a new infinite family of real flat ETFs. Another result establishes an equivalence between real flat ETFs and certain types of quasi-symmetric designs, resulting in a new infinite family of such designs

A Family of Quantum Stabilizer Codes Based on the Weyl Commutation Relations over a Finite Field

Using the Weyl commutation relations over a finite field we introduce a family of error-correcting quantum stabilizer codes based on a class of symmetric matrices over the finite field satisfying certain natural conditions. When the field is GF(2) the existence of a rich class of such symmetric matrices is demonstrated by a simple probabilistic argument depending on the Chernoff bound for i.i.d symmetric Bernoulli trials. If, in addition, these symmetric matrices are assumed to be circulant it is possible to obtain concrete examples by a computer program. The quantum codes thus obtained admit elegant encoding circuits.Comment: 16 pages, 2 figure

On the Exact Evaluation of Certain Instances of the Potts Partition Function by Quantum Computers

We present an efficient quantum algorithm for the exact evaluation of either the fully ferromagnetic or anti-ferromagnetic q-state Potts partition function Z for a family of graphs related to irreducible cyclic codes. This problem is related to the evaluation of the Jones and Tutte polynomials. We consider the connection between the weight enumerator polynomial from coding theory and Z and exploit the fact that there exists a quantum algorithm for efficiently estimating Gauss sums in order to obtain the weight enumerator for a certain class of linear codes. In this way we demonstrate that for a certain class of sparse graphs, which we call Irreducible Cyclic Cocycle Code (ICCC_\epsilon) graphs, quantum computers provide a polynomial speed up in the difference between the number of edges and vertices of the graph, and an exponential speed up in q, over the best classical algorithms known to date

On the Weight Distribution of Codes over Finite Rings

Let R > S be finite Frobenius rings for which there exists a trace map T from R onto S as left S modules. Let C:= {x -> T(ax + bf(x)) : a,b in R}. Then C is an S-linear subring-subcode of a left linear code over R. We consider functions f for which the homogeneous weight distribution of C can be computed. In particular, we give constructions of codes over integer modular rings and commutative local Frobenius that have small spectra.Comment: 18 p