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For each $n \in \mathbb{N} \cup \{\infty\}$, diagrammatic sets admit a model structure whose fibrant objects are the diagrammatic $(\infty, n)$- categories. They also support a notion of Gray product given by the Day convolution of a monoidal structure on their base category. The goal of this article is to show that the model structures are monoidal with respect to the Gray product. On the way to the result, we also prove that the Gray product of any cell and an equivalence is again an equivalence. Finally, we show that tensoring on the left or the right with the walking equivalence is a functorial cylinder for the model structures, and that the functor sending a diagrammatic set to its opposite is a Quillen self-equivalence.

The Gray configuration is a (27_3) configuration which typically is realized as the points and lines of the 3 x 3 x 3 integer lattice. It occurs as a member of an infinite family of configurations defined by Bouwer in 1972. Since their discovery, both the Gray configuration and its Levi graph (i.e., its point-line incidence graph) have been the subject of intensive study. Its automorphism group contains cyclic subgroups isomorphic to Z_3 and Z_9, so it is natural to ask whether the Gray configuration can be realized in the plane with any of the corresponding rotational symmetry. In this paper, we show that there are two distinct polycyclic realizations with Z_3 symmetry. In contrast, the only geometric polycyclic realization with straight lines and Z_9 symmetry is only a "weak" realization, with extra unwanted incidences (in particular, the realization is actually a (27_4) configuration).

Robust Gray codes were introduced by (Lolck and Pagh, SODA 2024). Informally, a robust Gray code is a (binary) Gray code $\mathcal{G}$ so that, given a noisy version of the encoding $\mathcal{G}(j)$ of an integer $j$, one can recover $\hat{j}$ that is close to $j$ (with high probability over the noise). Such codes have found applications in differential privacy. In this work, we present near-optimal constructions of robust Gray codes. In more detail, we construct a Gray code $\mathcal{G}$ of rate $1 - H_2(p) - \varepsilon$ that is efficiently encodable, and that is robust in the following sense. Supposed that $\mathcal{G}(j)$ is passed through the binary symmetric channel $\text{BSC}_p$ with cross-over probability $p$, to obtain $x$. We present an efficient decoding algorithm that, given $x$, returns an estimate $\hat{j}$ so that $|j - \hat{j}|$ is small with high probability.

We study skew-tolerant Gray codes, which are Gray codes in which changes in consecutive codewords occur in adjacent positions. We present the first construction of asymptotically non-vanishing skew-tolerant Gray codes, offering an exponential improvement over the known construction. We also provide linear-time encoding and decoding algorithms for our codes. Finally, we extend the definition to non-binary alphabets, and provide constructions of complete $m$-ary skew-tolerant Gray codes for every base $m\geq 3$.

A robust Gray code, formally introduced by (Lolck and Pagh, SODA 2024), is a Gray code that additionally has the property that, given a noisy version of the encoding of an integer $j$, it is possible to reconstruct $\hat{j}$ so that $|j - \hat{j}|$ is small with high probability. That work presented a transformation that transforms a binary code $C$ of rate $R$ to a robust Gray code with rate $Ω(R)$, where the constant in the $Ω(\cdot)$ can be at most $1/4$. We improve upon their construction by presenting a transformation from a (linear) binary code $C$ to a robust Gray code with similar robustness guarantees, but with rate that can approach $R/2$.

In this paper we present cartesian structure for symmetric Gray-monoidal double categories. To do this we first introduce locally cubical Gray categories, which are three-dimensional categorical structures analogous to classical, locally globular, Gray categories. The motivating example comprises double categories themselves, together with their functors, transformations, and modifications. A one-object locally cubical Gray category is a Gray-monoidal double category. Braiding, syllepsis, and symmetry for these is introduced in a manner analogous to that for 2-categories. Adding cartesian structure requires the introduction of doubly-lax functors of double categories to manage the order of copies. The resulting theory is algebraically rather complex, largely due to the bureaucracy of linearizing higher-dimensional boundary constraints. Fortunately, it has a relatively simple and compelling representation in the graphical calculus of surface diagrams, which we present.

To ensure differential privacy, one can reveal an integer fuzzily in two ways: (a) add some Laplace noise to the integer, or (b) encode the integer as a binary string and add iid BSC noise. The former is simple and natural while the latter is flexible and affordable, especially when one wants to reveal a sparse vector of integers. In this paper, we propose an implementation of (b) that achieves the capacity of the BSC with positive error exponents. Our implementation adds error-correcting functionality to Gray codes by mimicking how software updates back up the files that are getting updated ("coded Gray code"). In contrast, the old implementation of (b) interpolates between codewords of a black-box error-correcting code ("Grayed code").

Differential spatial modulation (DSM) was recently proposed to overcome the challenge of channel estimation in spatial modulation (SM). In this letter, we propose a gray code order of antenna index permutations for DSM. To facilitate the implementation, the well-known Trotter-Johnson ranking and unranking algorithms are adopted, which results in similar computational complexity to the existing DSM that uses the lexicographic order. The coding gain achieved by the proposed gray code order over the existing lexicographic order is also analyzed and verified via simulations, which reveals a maximum of about 1.2dB for the case of four transmit antennas. Based on the gray coding framework, we further propose a diversity-enhancing scheme named intersected gray (I-gray) code order for DSM, where the permutations of active antenna indices are selected directly from the odd (even) positions of the full permutations in the gray code order. From analysis and simulations, it is shown that the I-gray code order can harvests an additional diversity order at the expense of only one information bit loss for each transmission with respect to the gray code order.

We construct a (lax) Gray tensor product of $(\infty,2)$-categories and characterize it via a model-independent universal property. Namely, it is the unique monoidal biclosed structure on the $\infty$-category of $(\infty,2)$-categories which agrees with the classical Gray tensor product of strict 2-categories when restricted to the Gray cubes (i.e. the Gray tensor powers $[1]^{\otimes n}$ of the arrow category).

Gray models, which replace spectrally-resolved opacities with a wavelength independent mean opacity, are currently seeing wide and diverse application. In this brief review, we discuss both the history of gray techniques as well as recent applications of gray models, with an emphasis on planetary atmospheres. Methods and results for generating mean opacities are summarized. We present examples where gray radiative transfer tools are incorporated into three-dimensional atmospheric circulation models. Gray techniques are also useful for problems in comparative climatology, and we inter-compare results from several generalized gray models as applied to the computation of convective fluxes in planetary atmospheres. Finally, we provide examples where future progress can be made in the development of gray models.

The geometric and algebraic properties of Gray categories with duals are investigated by means of a diagrammatic calculus. The diagrams are three-dimensional stratifications of a cube, with regions, surfaces, lines and vertices labelled by Gray category data. These can be viewed as a generalisation of ribbon diagrams. The Gray categories present two types of duals, which are extended to functors of strict tricategories with natural isomorphisms, and correspond directly to symmetries of the diagrams. It is shown that these functors can be strictified so that the symmetries of a cube are realised exactly. A new condition on Gray categories with duals called the spatial condition is defined. A class of diagrams for which the evaluation for spatial Gray categories is invariant under homeomorphisms is exhibited. This relation between the geometry of the diagrams and structures in the Gray categories proves useful in computations and has potential applications in topological quantum field theory.

A complementary Gray code for binary n-tuples is one that, when all the tuples are complemented, is identical to itself; this is equivalent to the complement of the first half of the code being identical to the second half. We generalize the notion of complementary to q-ary n-tuples, fixed size combinations of an n-set and permutations and, in each case, construct complementary Gray codes. We relax, as weakly as possible, the notions of complementary to cases where necessary conditions for existence are violated and construct Gray codes within the weakened definitions: these include binary n-tuples when n is odd and Lee metric q-ary n-tuples when n is odd and q is even. Finally a lemma used in the construction for permutations offers the first known cyclic Gray code for the permutations of a particular family of multisets.

The paper presents some aspects of the (gray level) image binarization methods used in artificial vision systems. It is introduced a new approach of gray level image binarization for artificial vision systems dedicated to industrial automation temporal thresholding. In the first part of the paper are extracted some limitations of using the global optimum thresholding in gray level image binarization. In the second part of this paper are presented some aspects of the dynamic optimum thresholding method for gray level image binarization. Starting from classic methods of global and dynamic optimal thresholding of the gray level images in the next section are introduced the concepts of temporal histogram and temporal thresholding. In the final section are presented some practical aspects of the temporal thresholding method in artificial vision applications form the moving scene in robotic automation class; pointing out the influence of the acquisition frequency on the methods results.

This paper introduces an isometry between the modular rings $\Z_{2^s}$ and $\Z_{2^{s-1}}$ with respect to the homogeneous weights. Certain product of these maps gives Carlet's generalised Gray map and also Vega's Gray map. For $s=2$ this reduces to popular Gray map. Several interesting properties of these maps are studied. Towards the end we list several interesting problems to work on.

For any integer $n\geq 1$ a middle levels Gray code is a cyclic listing of all bitstrings of length $2n+1$ that have either $n$ or $n+1$ entries equal to 1 such that any two consecutive bitstrings in the list differ in exactly one bit. The question whether such a Gray code exists for every $n\geq 1$ has been the subject of intensive research during the last 30 years, and has been answered affirmatively only recently [T. Mütze. Proof of the middle levels conjecture. Proc. London Math. Soc., 112(4):677--713, 2016]. In this work we provide the first efficient algorithm to compute a middle levels Gray code. For a given bitstring, our algorithm computes the next $\ell$ bitstrings in the Gray code in time $\mathcal{O}(n\ell(1+\frac{n}{\ell}))$, which is $\mathcal{O}(n)$ on average per bitstring provided that $\ell=Ω(n)$.

A combinatorial Gray code for a set of combinatorial objects is a sequence of all combinatorial objects in the set so that each object is derived from the preceding object by changing a small part. In this paper we design a Gray code for ordered trees with n vertices such that each ordered tree is derived from the preceding ordered tree by removing a leaf then appending a leaf elsewhere. Thus the change is just remove-and-append a leaf, which is the minimum.

This paper develops some combinatorics of the lax Gray cylinder on the cells of Θ understood as a full subcategory of the category of strict ω-categories. More, we construct a span relating the Cartesian cylinder, the Gray cylinder, and the shift functor.

A Gray code is a listing structure for a set of combinatorial objects such that some consistent (usually minimal) change property is maintained throughout adjacent elements in the list. While Gray codes for m-ary strings have been considered in the past, we provide a new, simple Gray code for fixed-weight m-ary strings. In addition, we consider a relatively new type of Gray code known as overlap cycles and prove basic existence results concerning overlap cycles for fixed-weight and weight-range m-ary words.

For any integer $n\geq 1$ a middle levels Gray code is a cyclic listing of all $n$-element and $(n+1)$-element subsets of $\{1,2,\ldots,2n+1\}$ such that any two consecutive subsets differ in adding or removing a single element. The question whether such a Gray code exists for any $n\geq 1$ has been the subject of intensive research during the last 30 years, and has been answered affirmatively only recently [T. Mütze. Proof of the middle levels conjecture. Proc. London Math. Soc., 112(4):677--713, 2016]. In a follow-up paper [T. Mütze and J. Nummenpalo. An efficient algorithm for computing a middle levels Gray code. To appear in ACM Transactions on Algorithms, 2018] this existence proof was turned into an algorithm that computes each new set in the Gray code in time $\mathcal{O}(n)$ on average. In this work we present an algorithm for computing a middle levels Gray code in optimal time and space: each new set is generated in time $\mathcal{O}(1)$ on average, and the required space is $\mathcal{O}(n)$.