We extended the notion of the usual trace map of a field extension to finite-dimensional commutative algebras over a field and demonstrated its applications to coding theory. In particular, we provided a construction of subfield codes of codes defined over finite-dimensional GF(q)-algebras, thereby extending the work of Ding and Heng from fields to algebras. To the best of our knowledge, we are the first to determine the weight distribution of subfield codes of codes over algebras, and we show that subfield codes of C_D-codes over GF(2)[x]/ <x^3 - x > possess good parameters.
For n > 1, the n-th cyclotomic polynomial Q_n(x) is a divisor of x^n - 1 in GF(q)[x]. Hence, it is natural to study the parameters of the cyclic code C_n over GF(q) generated by Q_n(x). For every n > 1, we determine the parameters of the code C_n and one of its subcodes generated by (x - 1)Q_n(x), and show that these parameters are functions of n. We also prove that the distances of their Euclidean duals are functions of n, specifically 2^{omega(n)}. We further utilize these codes to construct several optimal families of cyclic LCD locally repairable codes.
Inspired by the work of Beelen et al. and Liu et al., we introduced a new family of codes, which we call Row–Column Twisted Reed–Solomon (RCTRS) codes. We provide sufficient conditions under which these codes are MDS and establish their existence. Furthermore, by examining their Schur squares, we prove that RCTRS codes are non-Reed–Solomon MDS codes. Finally, we show that RCTRS codes are not equivalent to column-twisted Reed–Solomon codes, thereby highlighting the novelty of our construction.
Recently, we introduced multi-twisted Goppa (MTG) codes and developed a decoding algorithm for the case of a single twist term, leading to a corresponding McEliece public-key encryption (PKE) scheme. While McEliece variants based on binary irreducible Goppa codes have been shown to be vulnerable to partial key-recovery attacks, we demonstrate that the MTG-based construction resists such attacks.