2023/03/12 by Arthur Bik, Bik, Arthur, Alessandro Neri +1
Computer Science · Engineering · #11T71 #14N07 #51E22 #94B05 #Advanced Wireless Communication Techniques #Coding theory and cryptography #Combinatorics (math.CO) #Commutative Algebra (math.AC) #FOS: Computer and information sciences #FOS: Mathematics #Information Theory (cs.IT) #graph theory and CDMA systems
paper · pdf · doi:10.48550/arxiv.2303.06745
openalex publication_date 2023/03/12 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Over fields of characteristic unequal to 2, we can identify symmetric matrices with homogeneous polynomials of degree 2. This allows us to view symmetric rank-metric codes as living inside the space of such polynomials. In this paper, we generalize the construction of symmetric Gabidulin codes to polynomials of degree d>2 over field of characteristic 0 or >d. To do so, we equip the space of homogeneous polynomials of degree d≥ 2 with the metric induced by the essential rank, which is the minimal number of linear forms needed to express a polynomial. We provide bounds on the minimal distance and dimension of the essential-rank metric codes we construct and provide an efficient decoding algorithm. Finally, we show how essential-rank metric codes can be seen as special instances of rank-metric codes and compare our construction to known rank-metric codes with the same parameters.