Generalization of low rank parity-check (LRPC) codes over the ring of integers modulo a positive integer

Following the work of Gaborit et al. (in: The international workshop on coding and cryptography (WCC 13), 2013) defining LRPC codes over finite fields, Renner et al. (in: IEEE international symposium on information theory, ISIT 2020, 2020) defined LRPC codes over the ring of integers modulo a prime power, inspired by the paper of Kamche and Mouaha (IEEE Trans Inf Theory 65(12):7718–7735, 2019) which explored rank metric codes over finite principal ideal rings. In this work, we successfully extend the work of Renner et al. by constructing LRPC codes over the ring $$\mathbb {Z}_{m}$$ Z m which is not a chain ring. We give a decoding algorithm and we study the failure probability of the decoder.

Determinants of some special matrices over commutative finite chain rings

Circulant matrices over finite fields and over commutative finite chain rings have been of interest due to their nice algebraic structures and wide applications. In many cases, such matrices over rings have a closed connection with diagonal matrices over their extension rings. In this paper, the determinants of diagonal and circulant matrices over commutative finite chain rings R with residue field 𝔽q are studied. The number of n × n diagonal matrices over R of determinant a is determined for all elements a in R and for all positive integers n. Subsequently, the enumeration of nonsingular n × n circulant matrices over R of determinant a is given for all units a in R and all positive integers n such that gcd(n, q) = 1. In some cases, the number of singular n × n circulant matrices over R with a fixed determinant is determined through the link between the rings of circulant matrices and diagonal matrices. As applications, a brief discussion on the determinants of diagonal and circulant matrices over commutative finite principal ideal rings is given. Finally, some open problems and conjectures are posted