On PIR schemes from nD-cyclic codes and the Schur product of some algebraic codes

Date

2026-1-22

Author

Karakaş, Burcu Ecem

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Private Information Retrieval (PIR) schemes aim to retrieve data from a database without revealing any details which data has been retrieved. A PIR scheme for coded storage systems with colluding servers gives a better PIR rate when the storage code and retrieval code have transitive automorphism groups. In this work, we study the transitivity of nD-cyclic codes and then PIR schemes from them together with several examples of nD-cyclic codes with better PIR rates. Then, we show the monomial equivalence between nD-cyclic codes and certain nD-constacyclic codes, which can be used as an alternative family of transitive codes. It has been shown that nD-cyclic codes are a particular type of quasi-cyclic codes. Another family of codes that is closely related to quasi-cyclic codes is additive cyclic codes. Using different representations, we investigate the Schur product of additive cyclic codes and explore their algebraic properties within this framework. In particular, we provide a study of an explicit polynomial generator matrix of the Schur product for additive cyclic codes of index 2 and 3, and then extend this study to arbitrary index, in terms of trace representation. Furthermore, we examine the distance properties of the Schur product of additive cyclic codes, extending BCH, HT and Hasse-Weil bounds known for cyclic codes.

Subject Keywords

Private Information Retrieval, Transitive codes, nD-cyclic codes, nD-constacyclic codes, Additive cyclic codes, Schur product, BCH bound, Hasse-Weil bound

URI

https://hdl.handle.net/11511/118376

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Graduate School of Applied Mathematics, Thesis