Designs, Codes and Cryptography
Construction of a (64, 2 ^{ 37}, 12) Codevia Galois Rings
Designs, Codes and Cryptography
The MAGMA algebra system I: the user language
Journal of Symbolic Computation - Special issue on computational algebra and number theory: proceedings of the first MAGMA conference
Finite-ring combinatorics and MacWilliams' equivalence theorem
Journal of Combinatorial Theory Series A
On arcs in projective Hjelmslev planes
Discrete Mathematics - Special issue on the 17th british combinatorial conference selected papers
Trace-Function on a Galois Ring in Coding Theory
AAECC-12 Proceedings of the 12th International Symposium on Applied Algebra, Algebraic Algorithms and Error-Correcting Codes
Ring geometries, two-weight codes, and strongly regular graphs
Designs, Codes and Cryptography
Cyclic codes over Z4, locator polynomials, and Newton's identities
IEEE Transactions on Information Theory
Cyclic codes and quadratic residue codes over Z4
IEEE Transactions on Information Theory
Gray isometries for finite chain rings and a nonlinear ternary (36, 312, 15) code
IEEE Transactions on Information Theory
Quasi-cyclic codes over Z4 and some new binary codes
IEEE Transactions on Information Theory
The Z4-linearity of Kerdock, Preparata, Goethals, and related codes
IEEE Transactions on Information Theory
Quaternary quadratic residue codes and unimodular lattices
IEEE Transactions on Information Theory
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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 $${\mathbb {Z}_4}$$ 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 $${\mathbb {Z}_4}$$ -preimages of the Kerdock codes and a related family of codes we will call Teichmüller 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, 27, 28)2, (60, 28, 28)2, (114, 28, 56)2, (372, 210, 184)2 and (1988, 212, 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, 29, 88)2, (244, 29, 120)2, (484, 210, 240)2 and (504, 46, 376)4 where no comparable (in the above sense) linear code is known (better-than-known-linear, BTKL).