Automorphism Groups and Isometries for Cyclic Orbit Codes

On the Automorphism Groups and Equivalence of Cyclic Combinatorial Objects

We determine the permutation groups that arise as the automorphism groups of cyclic combinatorial objects. As special cases we classify the automorphism groups of cyclic codes. We also give the permutations by which two cyclic combinatorial objects on $p^m$ elements are equivalent.

具有给定自同构的线性码的结构研究

On the Structure of the Linear Codes with a Given Automorphism

本文研究有限域上具m阶置换自同构的线性码结构，将其归为广义准循环码，给出准循环码等为自正交、自对偶或线性互补对偶码的充要条件。

Automorphism Ensemble Decoding of Quasi-Cyclic LDPC Codes by Breaking Graph Symmetries

We consider automorphism ensemble decoding (AED) of quasi-cyclic (QC) low-density parity-check (LDPC) codes. Belief propagation (BP) decoding on the conventional factor graph is equivariant to the quasi-cyclic automorphisms and therefore prevents gains by AED. However, by applying small modifications to the parity-check matrix at the receiver side, we can break the symmetry without changing the code at the transmitter. This way, we can leverage a gain in error-correcting performance using an ensemble of identical BP decoders, without increasing the worst-case decoding latency. The proposed met…

Non-standard linear recurring sequence subgroups and automorphisms of irreducible cyclic codes

Let \(\cU\) be the multiplicative group of order~\(n\) in the splitting field \(\bbF_{q^m}\) of \(x^n-1\) over the finite field \(\bbF_q\). Any map of the form \(x\rightarrow cx^t\) with \(c\in \cU\) and \(t=q^i\), \(0\leq i<m\), is \(\bbF_q\)-linear on~\(\bbF_{q^m}\) and fixes \(\cU\) set-wise; maps of this type will be called {\em standard\/}. Occasionally there are other, {\em non-standard\/} \(\bbF_q\)-linear maps on~\(\bbF_{q^m}\) fixing \(\cU\) set-wise, and in that case we say that the pair \((n, q)\) is {\em non-standard\/}. We show that an irreducible cyclic code of length~\(n\) over …

New Two-Stage Automorphism Group Decoders for Cyclic Codes in the Erasure Channel

Recently, error correcting codes in the erasure channel have drawn great attention for various applications such as distributed storage systems and wireless sensor networks, but many of their decoding algorithms are not practical because they have higher decoding complexity and longer delay. Thus, the automorphism group decoder (AGD) for cyclic codes in the erasure channel was introduced, which has good erasure decoding performance with low decoding complexity. In this paper, we propose new two-stage AGDs (TS-AGDs) for cyclic codes in the erasure channel by modifying the parity check matrix an…

Isometrically Self-dual Cyclic Codes

General isometries of cyclic codes, including multipliers and translations, are introduced; and isometrically self-dual cyclic codes are defined. In terms of Type-I duadic splittings given by multipliers and translations, a necessary and sufficient condition for the existence of isometrically self-dual cyclic codes is obtained. A program to construct isometrically self-dual cyclic codes is provided, and illustrated by several examples. In particular, a class of isometrically self-dual MDS cyclic codes, which are alternant codes from a class of generalized Reed-Solomon codes, is presented.