Related papers: Toric Codes, Multiplicative Structure and Decoding
We show a construction of a quantum ramp secret sharing scheme from a nested pair of linear codes. Necessary and sufficient conditions for qualified sets and forbidden sets are given in terms of combinatorial properties of nested linear…
The toric code can be constructed as a gauge theory of finite groups on oriented two dimensional lattices. Here we construct analogous models with the gauge fields belonging to groupoids, which are categories where every morphism has an…
We present a construction of subspace codes along with an efficient algorithm for list decoding from both insertions and deletions, handling an information-theoretically maximum fraction of these with polynomially small rate. Our…
This article presents new constructions of quantum error correcting Calderbank-Shor-Steane (CSS for short) codes. These codes are mainly obtained by Sloane's classical combinations of linear codes applied here to the case of self-orthogonal…
Quantum error correction is indispensable to achieving reliable quantum computation. When quantum information is encoded redundantly, a larger Hilbert space is constructed using multiple physical qubits, and the computation is performed…
An iterative decoding algorithm for convolutional codes is presented. It successively processes $N$ consecutive blocks of the received word in order to decode the first block. A bound is presented showing which error configurations can be…
We describe a new class of list decodable codes based on Galois extensions of function fields and present a list decoding algorithm. These codes are obtained as a result of folding the set of rational places of a function field using…
Minimal linear codes have significant applications in secret sharing schemes and secure two-party computation. There are several methods to construct linear codes, one of which is based on functions over finite fields. Recently, many…
Interleaved Reed-Solomon codes are applied in numerous data processing, data transmission, and data storage systems. They are generated by interleaving several codewords of ordinary Reed-Solomon codes. Usually, these codewords are decoded…
Long quantum codes using projective Reed-Muller codes are constructed. Projective Reed-Muller codes are evaluation codes obtained by evaluating homogeneous polynomials at the projective space. We obtain asymmetric and symmetric quantum…
Linear codes have diverse applications in secret sharing schemes, secure two-party computation, association schemes, strongly regular graphs, authentication codes and communication. There are a large number of linear codes with few weights…
Quantum error correction (QEC) is critical for scalable fault-tolerant quantum computing. Topological codes, such as the toric code, offer hardware-efficient architectures but their Tanner graphs contain many girth-4 cycles that degrade the…
Toric codes are error-correcting codes that are derived from toric varieties, which hold a unique correspondence to integral convex polytopes. In this paper, we focus on integral convex polytopes $P \subseteq \mathbb{R}^2$ and the toric…
Matrix multiplication over the real field constitutes a foundational operation in the training of deep learning models, serving as a computational cornerstone for both forward and backward propagation processes. However, the presence of…
We introduce a morphing procedure that can be used to generate new quantum codes from existing quantum codes. In particular, we morph the 15-qubit Reed-Muller code to obtain a $[\![10,1,2]\!]$ code that is the smallest known stabilizer code…
We show how the theory of affine geometries over the ring ${\mathbb Z}/\langle q - 1\rangle$ can be used to understand the properties of toric and generalized toric codes over ${\mathbb F}_q$. The minimum distance of these codes is strongly…
There are several methods for constructing secret sharing schemes, one of which is based on coding theory. Theoretically, every linear code can be used to construct secret sharing schemes. However, in general, determining the access…
We present an algorithm for error correction in topological codes that exploits modern machine learning techniques. Our decoder is constructed from a stochastic neural network called a Boltzmann machine, of the type extensively used in deep…
We investigate the geometrical structures of multipartite states based on construction of toric varieties. In particular, we describe pure quantum systems in terms of affine toric varieties and projective embedding of these varieties in…
We introduce a new class of circuits for constructing efficiently decodable error-correction codes, based on a recently discovered contractible tensor network. We perform an in-depth study of a particular example that can be thought of as…