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In this paper, we employ the linear systems representation of a convolutional code to develop a decoding algorithm for convolutional codes over the erasure channel. We study the decoding problem using the state space description and this…
The increasing interest in using quantum error correcting codes in practical devices has heightened the need for designing quantum error correcting codes that can correct against specialized errors, such as that of amplitude damping errors…
In this paper, we study algebraic geometry codes from curves over $\mathbb{F}_{q^\ell}$ through their virtual projections which are algebraic geometric codes over $\mathbb{F}_q$. We use the virtual projections to provide fractional decoding…
Algebraic-geometric codes on Garcia-Stichtenoth family of curves are used to construct the asymptotically good quantum codes.
This paper introduces sparse coding and dictionary learning for Symmetric Positive Definite (SPD) matrices, which are often used in machine learning, computer vision and related areas. Unlike traditional sparse coding schemes that work in…
The deletion channel is known to be a notoriously diffcult channel to design error-correction codes for. In spite of this difficulty, there are some beautiful code constructions which give some intuition about the channel and about what…
Non-binary codes correcting multiple deletions have recently attracted a lot of attention. In this work, we focus on multiplicity-free codes, a family of non-binary codes where all symbols are distinct. Our main contribution is a new…
The theory of error-correcting codes is concerned with constructing codes that optimize simultaneously transmission rate and relative minimum distance. These conflicting requirements determine an asymptotic bound, which is a continuous…
We study permutation groups of given minimal degree without the classical primitivity assumption. We provide sharp upper bounds on the order of a permutation group of minimal degree m and on the number of its elements of any given support.…
We propose a novel decoding approach for neural machine translation (NMT) based on continuous optimisation. We convert decoding - basically a discrete optimization problem - into a continuous optimization problem. The resulting constrained…
Color transfer is an image editing process that adjusts the colors of a picture to match a target picture's color theme. A natural color transfer not only matches the color styles but also prevents after-transfer artifacts due to image…
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…
Quantum secret-sharing and quantum error-correction schemes rely on multipartite decoding protocols, yet the non-local operations involved are challenging and sometimes infeasible. Here we construct a quantum secret-sharing protocol with a…
We describe a quantum error correction scheme aimed at protecting a flow of quantum information over long distance communication. It is largely inspired by the theory of classical convolutional codes which are used in similar circumstances…
Mitigating errors in computing and communication systems has seen a great deal of research since the beginning of the widespread use of these technologies. However, as we develop new methods to do computation or communication, we also need…
Cylindrical Algebraic Decomposition (CAD) is a key tool in computational algebraic geometry, best known as a procedure to enable Quantifier Elimination over real-closed fields. However, it has a worst case complexity doubly exponential in…
Sarvepalli and Klappenecker showed how classical one-point codes on the Hermitian curve can be used to construct quantum codes. Homma and Kim determined the parameters of a larger family of codes, the two-point codes. In quantum…
The Clifford defect is a rational number associated to the Weierstrass semigroup at a given point of an algebraic curve. It describes the error-correcting capability of the so-called Modified Algorithm for decoding the corresponding…
We introduce a generalisation of quantum error correction, relaxing the requirement that a code should identify and correct a set of physical errors on the Hilbert space of a quantum computer exactly, instead allowing recovery up to a…
Two mappings in a finite field, the Frobenius mapping and the cyclic shift mapping, are applied on lines in PG($n,p$) or codes in the Grassmannian, to form automorphisms groups in the Grassmanian and in its codes. These automorphisms are…