In the work [1], we developed a method to analyze the dimension of quadratic-curve-lifted Reed-Solomon codes and its asymptotic behavior. The method gives tighter bound on the dimension compared to the estimation given in [2], which defines a more general class of lifted Reed-Solomon codes with regard to curves of higher degree. 

In this work, the student is expected to extended the method in [1] to the general class of codes in [2] to analyse the dimension and its asymptotic behavior.

Security and privacy analysis on LRS codes or MR-LRCs

The explosion in the volumes of data being stored online has resulted in distributed storage `s transitioning to erasure coding based schemes. Local Reconstruction Codes (LRCs) have emerged as the codes of choice for these applications. These codes can correct a small number of erasures (which is the typical case) by accessing only a small number of remaining coordinates. A maximally recoverable (MR) LRC offers the best possible blend of such local and global fault tolerance, guaranteeing recovery from all erasure patterns which are information-theoretically correctable given the presence of local recovery groups. MR LRCs have received much attention recently, with many explicit constructions covering different regimes of parameters. 

In this work, the student is expected to understand and be able to explain the derivation of lower bound on the field size and present the constructions  given in [1] of MR-LRCs with a few global parities.

[1]S. Gopi, V. Guruswami and S. Yekhanin, "Maximally Recoverable LRCs: A Field Size Lower Bound and Constructions for Few Heavy Parities," in IEEE Transactions on Information Theory, vol. 66, no. 10, pp. 6066-6083, Oct. 2020, doi: 10.1109/TIT.2020.2990981.

Channel Coding

Information Theory

Coding Theory for Storage and Networks (Recommonded)

Local Reconstruction Codes (LRCs) allow for recovery from a small number of erasures in a local manner based on just a few other codeword symbols. They have emerged as the codes of choice for large scale distributed storage systems due to the very efficient repair of failed storage nodes in the typical scenario of a single or few nodes failing, while also offering fault tolerance against worst-case scenarios with more erasures. A maximally recoverable (MR) LRC offers the best possible blend of such local and global fault tolerance, guaranteeing recovery from all erasure patterns which are information-theoretically correctable given the presence of local recovery groups.

Skew polynomials, as a class of non-commutative polynomials, are gaining research interest for their rich applications in coding and complexity theory.

In this work, the student is expected to understand the theory of skew polynomials and MR-LRCs from [1] and present the construction from [2].

[1] Martínez-Peñas, Umberto, Mohannad Shehadeh, and Frank R. Kschischang. "Codes in the Sum-Rank Metric: Fundamentals and Applications." Foundations and Trends® in Communications and Information Theory 19.5 (2022): 814-1031. (Chapter 2-3)

[2] Gopi, Sivakanth, and Venkatesan Guruswami. "Improved maximally recoverable LRCs using skew polynomials." IEEE Transactions on Information Theory (2022).

Linear Algebra

Channel Coding

Coding Theory for Storage and Networks (recommended)