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Abstract The manipulation of 3D objects is becoming crucial for many applications, such as health, industry, or entertainment, to mention some. However, these 3D objects require substantial energy and different types of resources. With the goal of obtaining a simplified representation of a 3D object that can be easily managed, for example, for transmission, in some recent works, the authors associate low-density point clouds with a 3D object that simplifies the original 3D object. More precisely, given a 3D object in a polyhedral format, some authors associate a chain code and then use grammar-free context to obtain key points that give rise to several point clouds with different densities. In this work, we complete the cycle by developing a polyhedral reconstruction from an associated low-density point cloud and the chain code. The polyhedral reconstruction is crucial for handling 3D objects because it allows us to visualize them after they are efficiently compressed and transmitted. We apply our algorithms to well-known 3D objects in the literature. We use the Hausdorff and Chamfer distances to compare our results with the state-of-the-art proposals. We show how our proposed polyhedral reconstruction based on a helical chain code reconstructs a medical image represented or transmitted by slices into a 3D object in a polyhedral format, helping thus to mitigate and alleviate the management of 3D medical objects. The polyhedron that we propose provides better compression when compared with the original set of slices of a 3D medical object. 

Abstract The Generalized Hamming weights and their relative version, which generalize the minimum distance of a linear code, are relevant to numerous applications, including coding on the wire-tap channel of type II,t-resilient functions, bounding the cardinality of the output in list decoding algorithms, ramp secret sharing schemes, and quantum error correction. The generalized Hamming weights have been determined for some families of codes, including Cartesian codes and Hermitian one-point codes. In this paper, we determine the generalized Hamming weights of decreasing norm-trace codes, which are linear codes defined by evaluating sets of monomials that are closed under divisibility on the rational points of the extended norm-trace curve given by$$x^{u} = y^{q^{s - 1}} + y^{q^{s - 2}} + \cdots + y$$ $x^u=y^{q^{s-1}}+y^{q^{s-2}}+\cdots +y$over the finite field of cardinality$$q^s$$ $q^s$, whereuis a positive divisor of$$\frac{q^s - 1}{q - 1}$$ $\frac{q^s-1}{q-1}$. As a particular case, we obtain the weight hierarchy of one-point norm-trace codes and recover the result of Barbero and Munuera (2001) giving the weight hierarchy of one-point Hermitian codes. We also study the relative generalized Hamming weights for these codes and use them to construct impure quantum codes with excellent parameters. 

Abstract CSS-T codes were recently introduced as quantum error-correcting codes that respect a transversal gate. A CSS-T code depends on a CSS-T pair, which is a pair of binary codes$$(C_1, C_2)$$ $(C_1,C_2)$such that$$C_1$$ $C_1$contains$$C_2$$ $C_2$,$$C_2$$ $C_2$is even, and the shortening of the dual of$$C_1$$ $C_1$with respect to the support of each codeword of$$C_2$$ $C_2$is self-dual. In this paper, we give new conditions to guarantee that a pair of binary codes$$(C_1, C_2)$$ $(C_1,C_2)$is a CSS-T pair. We define the poset of CSS-T pairs and determine the minimal and maximal elements of the poset. We provide a propagation rule for nondegenerate CSS-T codes. We apply some main results to Reed–Muller, cyclic and extended cyclic codes. We characterize CSS-T pairs of cyclic codes in terms of the defining cyclotomic cosets. We find cyclic and extended cyclic codes to obtain quantum codes with better parameters than those in the literature. 

Codes over rings, especially over Galois rings, have been extensively studied for nearly three decades due to their similarity to linear codes over finite fields. A distributed storage system uses a linear code to encode a large file across several nodes. If one of the nodes fails, a linear exact repair scheme efficiently recovers the failed node by accessing and downloading data from the rest of the servers of the storage system. In this paper, we develop a linear repair scheme for free maximum distance separable codes, which coincide with free maximum distance with respect to the rank codes over Galois rings. In particular, we give a linear repair scheme for full-length Reed–Solomon codes over a Galois ring.