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In this paper, we study mixed alphabet codes over two alphabets Z 2 , and its extension Z 2 [ u ] = Z 2 + u Z 2 , u 2 = 0. We define new Gray maps, using which we study cyclic codes and obtain the properties of their generators. Further, we construct constacyclic codes, and linear codes, and observe the useful properties of their generators. Finally, we present some examples of cyclic codes and binary linear codes. [ABSTRACT FROM AUTHOR]

Quadrics are important in finite geometry and can be used to construct binary codes. In this paper, we first define an incidence matrix M based on points and non-degenerate quadrics in the classical projective space PG (n − 1 , q) , where q is a prime power. As a consequence, we establish a binary code C (M) with the generator matrix M and determine the dimension of C (M) when q and n are both odd. In particular, we study the minimum distances of C (M) and C ⊥ (M) in PG (2 , q) and give their upper bounds. [ABSTRACT FROM AUTHOR]

Let X be an n-element point set in the k-dimensional unit cube [ 0 , 1 ] k where k ≥ 2 . According to an old result of Bollobás and Meir (Oper Res Lett 11:19–21, 1992) , there exists a cycle (tour) x 1 , x 2 , ... , x n through the n points, such that ∑ i = 1 n | x i - x i + 1 | k 1 / k ≤ c k , where | x - y | is the Euclidean distance between x and y, and c k is an absolute constant that depends only on k, where x n + 1 ≡ x 1 . From the other direction, for every k ≥ 2 and n ≥ 2 , there exist n points in [ 0 , 1 ] k , such that their shortest tour satisfies ∑ i = 1 n | x i - x i + 1 | k 1 / k = 2 1 / k · k . For the plane, the best constant is c 2 = 2 and this is the only exact value known. Bollobás and Meir showed that one can take c k = 9 2 3 1 / k · k for every k ≥ 3 and conjectured that the best constant is c k = 2 1 / k · k , for every k ≥ 2 . Here we significantly improve the upper bound and show that one can take c k = 3 5 2 3 1 / k · k or c k = 2.91 k (1 + o k (1)) . Our bounds are constructive. We also show that c 3 ≥ 2 7 / 6 , which disproves the conjecture for k = 3 . Connections to matching problems, power assignment problems, related problems, including algorithms, are discussed in this context. A slightly revised version of the Bollobás–Meir conjecture is proposed. [ABSTRACT FROM AUTHOR]

Quadrics are important in finite geometry and can be used to construct binary codes. In this paper, we first define an incidence matrix $ M $ based on points and non-degenerate quadrics in the classical projective space PG$ (n-1, q) $, where $ q $ is a prime power. As a consequence, we establish a binary code $ C(M) $ with the generator matrix $ M $ and determine the dimension of $ C(M) $ when $ q $ and $ n $ are both odd. In particular, we study the minimum distances of $ C(M) $ and $ C^{\perp}(M) $ in PG$ (2, q) $ and give their upper bounds.

Present manuscript designs and analyzes numerically all-optical binary-to-gray code (BTGC) converter utilizing silicon microring resonator. A waveguide-based silicon microring resonator has been employed to achieve optical switching under low-power conditions using the two-photon absorption effect. Gray code (GC) is a binary numerical system in which two consecutive codes distinguished by only one bit. The GC is critical in optics communication because it prevents spurious output from optical switches and facilitates error correction in optical communications. MATLAB is used to design and analyze the architecture at almost 260 Gbps operational speed. The faster response times and compact design of the demonstrated circuits make them especially useful for optical communication systems. Performance indicating factors evaluated from MATLAB results and analyzed. Design parameters that are optimized have been chosen in order to construct the model practically. [ABSTRACT FROM AUTHOR]

Homologous vulnerability detection is an important aspect of computer security. It has several key problems, including discriminating structurally complex functions, supporting cross-architecture programs, distinguishing false positives, etc. Non-homologous functions with similar control flow graph structures are easily misjudged, which decreases discrimination accuracy. The vectors generated by instruction-embedding models contain architectural features, which increases the distance between homologous function vectors and leads to misclassification. In this paper, we propose a novel heterogeneous graph embedding (HGE) binary vulnerability homology discrimination (BVHD) method. HGE is used to aggregate basic block features to generate function representations, perform different transformations according to control flow and data flow, and improve the discrimination of non-homologous functions to increase discrimination accuracy. A novel multi-architecture instruction-embedding model is proposed for abstracting common semantic features and weakening the interference of architectural features to avoid misclassification. The experimental results show that the proposed method achieves state-of-the-art results in homologous function discrimination, and the upgrade is significant for complex structure functions. [ABSTRACT FROM AUTHOR]