Tag - Error-correcting codes

Quoos Luciane: Semigroups and algebraic geometry codes

6th November, 2023

A linear code is a vector subspace of 𝔽qn, where 𝔽q is a finite field with q elements. The family of linear error-correcting codes are specially important when one is attempting to transmit messages across a noisy communication channel. Data can be corrupted in transmission or storage by a variety of undesirable phenomenon, such as radio interference, electrical noise, scratch, etc.. It is useful to have a way to detect and correct such data corruption. An error-correcting code can correct more errors larger is its minimum distance. This course aims to introduce a family of error-correcting codes, the Algebraic Geometry Codes, and show how to use the theory of semigroups to improve the minimum distance of the code. This construction of codes make use of a function field in one variable over a finite field. We will show how the local information in one or two rational places, the knowledge of the semigroup in these places, can be used to improve the minimum distance of the code.

Shruti Puri: Good qubits and good codes

26th October, 2023

In this talk I will discuss some criteria for good codes and good qubits for quantum error correction. A good qubit has a highly structured noise channel and good codes are the ones that have special underlying symmetries with respect to this channel. For fault-tolerance it is also necessary that we are able to implement the code with physical operations that preserve the symmetries. I will highlight this deep connection between what makes a good qubit and good code using Rydberg atom qubits and surface codes.

Victor V. Albert: Something for everybody: modern quantum tools for bosonic systems

28th April, 2023

I overview recent extensions of state-of-the-art discrete-variable (DV) tomographic, error-correction, and cryptographic protocols to continuous-variable (CV) systems, including: (1) a theory of appropriately defined CV state designs, and their applications to design-based CV shadow tomography; (2) a cryptographic protocol utilizing squeezed states whose proof of security is based on a CV extension of DV monogamy-of-entanglement games; (3) sample efficiency of homodyne and photon-number-resolving tomography obtained via recasting said protocols in terms of shadow tomography; and (4) a new class of quantum spherical codes inspired by classical spherical codes.

María Chara: An introduction to algebraic geometry codes

6th December, 2021

Error-correcting codes play an important role in many areas of science and engineering, as they safeguard the integrity of data against the adverse effects of noise in communication and storage. On the most basic level, good error-correcting codes are able to both transmit data efficiently and correct a large number of errors relative to their length. As observed by V. D. Goppa in 1975, one can use algebraic function fields over 𝔽q to construct a large class of interesting codes. Properties of these codes are closely related to properties of the corresponding function field, and the Riemman-Roch Theorem provides estimates, sharps in many cases, for their main parameters. In this short course we will study Goppa's construction of codes by means of an algebraic function field after a brief introduction of the theory of error-correcting codes, some classical bounds for the parameters of these codes and their detection and error-correction capabilities.