This paper focuses on the construction and analysis of codes over the Frobenius ring T. By exploiting the idempotent structure of the ring, codes over T are decomposed into direct sums of component codes over a related local Frobenius non-chain ring. A gray map is constructed and shown to be bijective and distance-preserving, allowing code parameters to be determined from their gray images. Linear codes and cyclic codes over the considered ring are completely characterized, and explicit descriptions of their generating sets and cardinalities are obtained. The duals of cyclic codes are also examined, and the gray map is shown to preserve orthogonality and self-duality. Well-parameterized codes were obtained over the ring T.
LCD codes over finite commutative chain rings are known to be free. However, if the ring is not indecomposable, there always exists an LCD ideal that is projective but not free. We prove that all two-sided LCD codes are projective. We generalise the characterisation of finite commutative Frobenius rings by means of the...
The Berger-Charpin conjecture predicts that the permutation automorphism group of a cyclic code is generally the $q$-affine group, which is generated by the shift and the Frobenius multiplier on the index set. The word {\em generally} is not made precise in the literature, and the goal of this series of papers is to st...
In this paper, we present a method for constructing linear codes derived from book graphs. Applying such a method, we construct binary codes that have the greatest possible minimum distance for their length and dimension by means of the $2$-sum of two equal book graphs. We then utilize matroid and graph theory together...
A. S. Barbosa, Franklin de L. Marquezino, G. L. La Guardia· 0 citations
We give a proof of the existence of two-sided product expanding codes which, unlike the earlier result of Kalachev and Panteleev (FOCS, 2025), does not rely on explicit constructions of asymptotically optimal locally testable codes ($c^3$-LTCs). For every fixed number of component codes and dimensions whose rates are b...
Eshan Chattopadhyay, Noam Ringach, Nicholas Spooner· 0 citations
Linearized permutation polynomials over finite fields have important applications in coding theory, cryptography, and combinatorics. In this paper, we give an explicit description of the cycle structure of a general linearized permutation polynomial in terms of the Jordan blocks of its Dickson matrix, thereby providing...
