New ring-linear codes from dualization in projective Hjelmslev geometries
In this article, several new constructions for ring-linear codes are given. The class of base rings are the Galois rings of characteristic 4, which include as its smallest and most important member. Associated with these rings are the Hjelmslev geometries, and the central tool for the construction is geometric dualization. Applying it to the -preimages of the Kerdock codes and a related family of codes we will call Teichmuller codes, we get two new infinite series of codes and compute their symmetrized weight enumerators. In some cases, residuals of the original code give further interesting codes. The generalized Gray map translates our codes into ordinary, generally non-linear codes in the Hamming space. The obtained parameters include (58, 2(7), 28)(2), (60, 2(8), 28)(2), (114, 2(8), 56)(2), (372, 2(10), 184)(2) and (1988, 2(12), 992)(2) which provably have higher minimum distance than any linear code of equal length and cardinality over an alphabet of the same size (better-than-linear, BTL), as well as (180, 2(9), 88)(2), (244, 2(9), 120)(2), (484, 2(10), 240)(2) and (504, 4(6), 376)(4) where no comparable (in the above sense) linear code is known (better-than-known-linear, BTKL).
ring-linear code hjelmslev geometry galois ring kerdock code gray map lee weight homogeneous weight