Construction of self-dual codes over finite rings Zpm
Journal of Combinatorial Theory Series A
Journal of Combinatorial Theory Series A
Cyclic and negacyclic codes over the Galois ring GR(p2,m)
Discrete Applied Mathematics
A generalization of the binary Preparata code
Discrete Applied Mathematics - Special issue: Coding and cryptography
Repeated-root cyclic and negacyclic codes over a finite chain ring
Discrete Applied Mathematics - Special issue: Coding and cryptography
Symmetric bilinear forms over finite fields of even characteristic
Journal of Combinatorial Theory Series A
The 2-distance coloring of the Cartesian product of cycles using optimal Lee codes
Discrete Applied Mathematics
An invariant for quadratic forms valued in Galois Rings of characteristic 4
Finite Fields and Their Applications
3-Designs from all Z4-Goethals-like codes with block size 7 and 8
Finite Fields and Their Applications
On the decomposition of self-dual codes over F2+uF2 with an automorphism of odd prime order
Finite Fields and Their Applications
Codes over Rk, Gray maps and their binary images
Finite Fields and Their Applications
Witt index for Galois Ring valued quadratic forms
Finite Fields and Their Applications
Self-dual codes over commutative Frobenius rings
Finite Fields and Their Applications
Nordstrom--Robinson code and A7-geometry
Finite Fields and Their Applications
A class of constacyclic codes over Zpm
Finite Fields and Their Applications
A note on cyclic codes over GR (p2,m) of length pk
Finite Fields and Their Applications
Some families of Z4-cyclic codes
Finite Fields and Their Applications
On the equivalence of codes over rings and modules
Finite Fields and Their Applications
Homogeneous weights and exponential sums
Finite Fields and Their Applications
On maximal arcs in projective Hjelmslev planes over chain rings of even characteristic
Finite Fields and Their Applications
Constructions of Low Rank Relative Difference Sets in 2-Groups Using Galois Rings
Finite Fields and Their Applications
Codes over p-adic Numbers and Finite Rings Invariant under the Full Affine Group
Finite Fields and Their Applications
Orthogonality Matrices for Modules over Finite Frobenius Rings and MacWilliams' Equivalence Theorem
Finite Fields and Their Applications
Factoring polynomials over Z4 and over certain Galois rings
Finite Fields and Their Applications
On the classification and enumeration of self-dual codes
Finite Fields and Their Applications
q-ary Bent Functions Constructed from Chain Rings
Finite Fields and Their Applications
New ring-linear codes from dualization in projective Hjelmslev geometries
Designs, Codes and Cryptography
Characteristics of invariant weights related to code equivalence over rings
Designs, Codes and Cryptography
Mass formula and structure of self-dual codes over $${{\bf Z}_{2^s}}$$
Designs, Codes and Cryptography
A remark on symplectic semifield planes and Z4-linear codes
Designs, Codes and Cryptography
New results on two hypercube coloring problems
Discrete Applied Mathematics
Linking systems in nonelementary abelian groups
Journal of Combinatorial Theory Series A
On self-dual cyclic codes over finite chain rings
Designs, Codes and Cryptography
Matrix product codes over finite commutative Frobenius rings
Designs, Codes and Cryptography
Hi-index | 754.84 |
Certain notorious nonlinear binary codes contain more codewords than any known linear code. These include the codes constructed by Nordstrom-Robinson (1967), Kerdock (1972), Preparata (1968), Goethals (1974), and Delsarte-Goethals (1975). It is shown here that all these codes can be very simply constructed as binary images under the Gray map of linear codes over Z4, the integers mod 4 (although this requires a slight modification of the Preparata and Goethals codes). The construction implies that all these binary codes are distance invariant. Duality in the Z4 domain implies that the binary images have dual weight distributions. The Kerdock and “Preparata” codes are duals over Z4-and the Nordstrom-Robinson code is self-dual-which explains why their weight distributions are dual to each other. The Kerdock and “Preparata” codes are Z4-analogues of first-order Reed-Muller and extended Hamming codes, respectively. All these codes are extended cyclic codes over Z4, which greatly simplifies encoding and decoding. An algebraic hard-decision decoding algorithm is given for the “Preparata” code and a Hadamard-transform soft-decision decoding algorithm for the I(Kerdock code. Binary first- and second-order Reed-Muller codes are also linear over Z4 , but extended Hamming codes of length n⩾32 and the Golay code are not. Using Z4-linearity, a new family of distance regular graphs are constructed on the cosets of the “Preparata” code