On Z4-linear Preparata-like and Kerdock-like codes

Designs, Codes and Cryptography, 2010

A code C is Z 2 Z 4 -additive if the set of coordinates can be partitioned into two subsets X and Y such that the punctured code of C by deleting the coordinates outside X (respectively, Y ) is a binary linear code (respectively, a quaternary linear code). In this paper Z 2 Z 4 -additive codes are studied. Their corresponding binary images, via the Gray map, are Z 2 Z 4 -linear codes, which seem to be a very distinguished class of binary group codes.

1999

arXiv: Information Theory, 2016

In this paper, we study a relative two-weight $\mathbb{Z}_2 \mathbb{Z}_4$-additive codes. It is shown that the Gray image of a two-distance $\mathbb{Z}_2 \mathbb{Z}_4$-additive code is a binary two-distance code and that the Gray image of a relative two-weight $\mathbb{Z}_2 \mathbb{Z}_4$-additive code, with nontrivial binary part, is a linear binary relative two-weight code. The structure of relative two-weight $\mathbb{Z}_2 \mathbb{Z}_4$-additive codes are described. Finally, we discussed permutation automorphism group of a $\mathbb{Z}_2 \mathbb{Z}_4$-additive codes.

IEEE Transactions on Information Theory, 1994

Certain notorious nonlinear binary codes contain more codewords than any known linear code. These include the codes constructed by Nordstrom-Robinson , Kerdock, Preparata, Goethals, and Delsarte-Goethals. It is shown here that all these codes can be very simply constructed as binary images under the Gray map of linear codes over Z 4 , 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 Z 4 domain implies that the binary images have dual weight distributions. The Kerdock and 'Preparata' codes are duals over Z 4-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 Z 4-analogues of first-order Reed-Muller and extended Hamming codes, respectively. All these codes are extended cyclic codes over Z 4 , 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 Kerdock code. Binary first-and second-order Reed-Muller codes are also linear over Z 4 , but extended Hamming codes of length n ≥ 32 and the Golay code are not. Using Z 4-linearity, a new family of distance regular graphs are constructed on the cosets of the 'Preparata' code.

Dcc, 2010

In this work, we investigate linear codes over the ring F 2 + uF 2 + vF 2 + uvF 2 . We first analyze the structure of the ring and then define linear codes over this ring which turns out to be a ring that is not finite chain or principal ideal contrary to the rings that have hitherto been studied in coding theory. Lee weights and Gray maps for these codes are defined by extending on those introduced in works such as Betsumiya et al. (Discret Math 275:43-65, 2004) and Dougherty et al. (IEEE Trans Inf 45:32-45, 1999). We then characterize the F 2 + uF 2 + vF 2 + uvF 2 -linearity of binary codes under the Gray map and give a main class of binary codes as an example of F 2 + uF 2 + vF 2 + uvF 2 -linear codes. The duals and the complete weight enumerators for F 2 + uF 2 + vF 2 + uvF 2 -linear codes are also defined after which MacWilliams-like identities for complete and Lee weight enumerators as well as for the ideal decompositions of linear codes over F 2 + uF 2 + vF 2 + uvF 2 are obtained.

We study odd and even Z 2 Z 4 formally self-dual codes. The images of these codes are binary codes whose weight enumerators are that of a formally self-dual code but may not be linear. Three constructions are given for formally self-dual codes and existence theorems are given for codes of each type defined in the paper.

European Journal of Combinatorics, 2004

We determine the parameters of the optimal additive quaternary codes of length at most 12 over Z 2 × Z 2 . Equivalently, we determine how many lines one can pick in a binary projective space such that any t are independent. Or again, how many lines one can pick in a binary projective space such that no hyperplane contains more than m of them.

4 is called cyclic code if the set of coordinates can be partitioned into two subsets, the set of Z 2 and the set of Z 4 coordinates, such that any cyclic shift of the coordinates of both subsets leaves invariant the code. These codes can be identified as submodules of the . The parameters of a Z 2 Z 4 -additive cyclic code are stated in terms of the degrees of the generator polynomials of the code. The generator polynomials of the dual code of a Z 2 Z 4 -additive cyclic code are determined in terms of the generator polynomials of the code C.

2021

In this paper we study the structure and properties of additive right and left polycyclic codes induced by a binary vector a in 𝔽_2^n. We find the generator polynomials and the cardinality of these codes. We also study different duals for these codes. In particular, we show that if C is a right polycyclic code induced by a vector a∈𝔽_2^n, then the Hermitian dual of C is a sequential code induced by a. As an application of these codes, we present examples of additive right polycyclic codes over 𝔽_4 with more codewords than comparable optimal linear codes as well as optimal binary linear codes and optimal quantum codes obtained from additive right polycyclic codes over 𝔽_4.

2010

A code C is Z 2 Z 4-additive if the set of coordinates can be partitioned into two subsets X and Y such that the punctured code of C by deleting the coordinates outside X (respectively, Y) is a binary linear code (respectively, a quaternary linear code). In this paper Z 2 Z 4-additive codes are studied. Their corresponding binary images, via the Gray map, are Z 2 Z 4-linear codes, which seem to be a very distinguished class of binary group codes. As for binary and quaternary linear codes, for these codes the fundamental parameters are found and standard forms for generator and parity check matrices are given. For this, the appropriate inner product is deduced and the concept of duality for Z 2 Z 4-additive codes is defined. Moreover, the parameters of the dual codes are computed. Finally, some conditions for self-duality of Z 2 Z 4-additive codes are given.