Partial Permutation Decoding for Several Families of Linear and ${\mathbb{Z}}_4$ -Linear CodesPartial Permutation Decoding for Several Families of Linear and ${\mathbb{Z}}_4$ -Linear Codes

Roland D. Barrolleta, Mercè Villanueva

A general criterion to obtain s-PD-sets of minimum size s + 1 for partial permutation decoding, which enable correction up to s errors, for systematic codes over a finite field Fq and Z4-linear codes is provided. We show how this technique can be easily applied to linear cyclic codes over Fq, Z4-linear codes which are the Gray map image of a quaternary linear cyclic code, and some related codes such as quasi-cyclic codes. Furthermore, specific results for some linear and nonlinear binary codes, including simplex, Kerdock, Delsarte-Goethals, and extended dualized Kerdock codes are given. Finally, applying this technique, new s-PD-sets of size s + 1 for Z4-linear Hadamard codes of type 2γ4δ, for all δ ≥ 4 and 1δ- 3; and for Z4-linear simplex codes of type 4m, for all m ≥ 2 and 1m+1- 3, are also provided.A general criterion to obtain s-PD-sets of minimum size s + 1 for partial permutation decoding, which enable correction up to s errors, for systematic codes over a finite field Fq and Z4-linear codes is provided. We show how this technique can be easily applied to linear cyclic codes over Fq, Z4-linear codes which are the Gray map image of a quaternary linear cyclic code, and some related codes such as quasi-cyclic codes. Furthermore, specific results for some linear and nonlinear binary codes, including simplex, Kerdock, Delsarte-Goethals, and extended dualized Kerdock codes are given. Finally, applying this technique, new s-PD-sets of size s + 1 for Z4-linear Hadamard codes of type 2γ4δ, for all δ ≥ 4 and 1δ- 3; and for Z4-linear simplex codes of type 4m, for all m ≥ 2 and 1m+1- 3, are also provided.

IEEE Transactions on Information TheoryMay 24, 20188 citations

Lineare Codes

Anton Betten, Harald Fripertinger, Adalbert Kerber +2 more

Springer eBooksJan 1, 19980 citations

Linear Codes

Toyokazu Hiramatsu, Günter Köhler

Springer eBooksJan 1, 20030 citations

A New Method of Constructing Binary Quantum Codes From Arbitrary Quaternary Linear Codes

Junli Wang, Ruihu Li, Jingjie Lv +1 more

In this letter, a method of constructing Hermitian dual containing quaternary linear codes from general quaternary linear codes is proposed. Based on this result, a construction method of binary linear quantum codes is obtained. Using three cyclic codes and one linear code over F4that are not Hermitian dual containing, we construct four record breaking binary linear quantum codes. Furthermore, ten new record breaking binary additive quantum codes are derived from these four linear quantum codes by propagation rule.In this letter, a method of constructing Hermitian dual containing quaternary linear codes from general quaternary linear codes is proposed. Based on this result, a construction method of binary linear quantum codes is obtained. Using three cyclic codes and one linear code over F4that are not Hermitian dual containing, we construct four record breaking binary linear quantum codes. Furthermore, ten new record breaking binary additive quantum codes are derived from these four linear quantum codes by propagation rule.

IEEE Communications LettersDec 25, 20193 citations

Linear Block Codes

Giuliano G. La Guardia

Quantum science and technologyJan 1, 20200 citations

Binary linear codes

Jürgen Bierbrauer

A binary code is linear if it is closed under addition. It is a basic fact from linear algebra that each linear code has a basis. Linear algebra applies to arbitrary fields. There is a general tendency to restrict attention to linear codes. One reason is that these are much easier to describe and to work with than codes in general. For a binary linear code, the minimum distance equals the minimum of the weights of nonzero codewords. Binary linear codes are particularly handy for block coding. In block coding the length of bitstrings is increased from k to n. That means that k/n of the bits sent through the channel represent pure information, whereas the rest is redundancy that we have cleverly introduced for the purposes of error correction.A binary code is linear if it is closed under addition. It is a basic fact from linear algebra that each linear code has a basis. Linear algebra applies to arbitrary fields. There is a general tendency to restrict attention to linear codes. One reason is that these are much easier to describe and to work with than codes in general. For a binary linear code, the minimum distance equals the minimum of the weights of nonzero codewords. Binary linear codes are particularly handy for block coding. In block coding the length of bitstrings is increased from k to n. That means that k/n of the bits sent through the channel represent pure information, whereas the rest is redundancy that we have cleverly introduced for the purposes of error correction.

Chapman & Hall/CRC eBooksOct 12, 20160 citations

Partial Permutation Decoding for Several Families of Linear and

{\mathbb{Z}}_4

-Linear Codes