A double Cyclic code (or DC code) of length n = k + l over Z2 is a binary linear code, where any Cyclic shift of the first k coordinates and the last l coordinates of a codeword is also a codeword. In this paper, we study the relationship between separability and self-duality of these codes. Also, we obtain the shadow code by determining the generator polynomials of the doubly even subcode of the self-dual code.

In this paper, we classify the skew Cyclic codes over Fp + vFp + v2Fp, where p is a prime number and v3 = v. Each skew Cyclic code is a Fp+vFp+v2Fp-submodule of the (Fp+vFp+v2Fp)[x; ϴ ], where v3 = v and ϴ (v) = − v. Also, we give an explicit forms for the generator of these codes. Moreover, an algorithm of encoding and decoding for these codes is presented.

An algebraic construction for constant dimension subspace codes is called orbit code. It arises as the orbits under the action of a subgroup of the general linear group on subspaces in an ambient space. In particular orbit codes of a Singer subgroup of the general linear group has investigated recently. In this paper, we consider the normalizer of a Singer subgroup of the general linear group and its orbit codes. Several properties of these codes are considered.

Abstract S. Kim et al. have been analyzed the girth of some algebraically structured quasi-Cyclic (QC) low-density parity-check (LDPC) codes, i.e. Tanner (3; 5) of length 5p, where p is a prime of the form 15m+1. In this paper, by extension this method to Tanner (3; 7) codes of length 7p, where p is a prime of the form 21m+1, the girth values of Tanner (3; 7) codes will be derived. As an advantage, the rate of Tanner (3; 7) codes is about 0: 17 more than the rate of Tanner (3; 5) codes.

In [30] a class of quasi-Cyclic LDPC codes has been proposed, whose information rate is 1/2. We generalize that construction to arbitrary rates l-1/l and we provide a Grobner basis for their dual codes, in some generic cases.

In this work, we study a class of skew Cyclic codes over the ring $R:=\mathbb{Z}_4+v\mathbb{Z}_4,$ where $v^2=v,$ with an automorphism $\theta$ and a derivation $\Delta_\theta,$ namely codes as modules over a skew polynomial ring $R[x;\theta,\Delta_{\theta}],$ whose multiplication is defined using an automorphism $\theta$ and a derivation $\Delta_{\theta}.$ We investigate the structures of a skew polynomial ring $R[x;\theta,\Delta_{\theta}].$ We define $\Delta_{\theta}$-Cyclic codes as a generalization of the notion of Cyclic codes. The properties of $\Delta_{\theta}$-Cyclic codes as well as dual $\Delta_{\theta}$-Cyclic codes are derived. As an application, some new linear codes over $\mathbb{Z}_4$ with good parameters are obtained by Plotkin sum construction, also via a Gray map as well as residue and torsion codes of these codes.

Background: Anti-Cyclic citrullinated peptide (CCP) antibodies are highly specific and predictive for rheumatoid arthritis (RA). Recently, existence of anti-CCP in several other inflammatory conditions from psoriatic arthritis to neurodegenerative disorder like Alzheimer’s disease (AD) has been reported. We compared the level of anti-CCP titer between the patients with mild cognitive impairment (MCI) and Alzheimer’s disease; then, we compared it between the patients with mild rheumatoid arthritis and Alzheimer’s disease.Methods: In this cross-sectional study, 40 patients with rheumatoid arthritis, 29 patients with Alzheimer’s disease, 30 patients with mild cognitive impairment and 30 healthy controls were enrolled. Anti-CCP titer was measured by solid phase enzyme immunoassay kit. Statistical analysis was performed using one-way analysis of variance and post-hoc Tukey tests.Findings: Level of anti-CCP titer (mean±SD) was higher significantly in patients with Alzheimer’ s disease (13.6±1.6) was higher than healthy subjects (4.2±1.1) and patients with mild cognitive impairment (4.8±1.3) (P=0.0100 for both). Anti-CCP serum level in 32.1% of patients with Alzheimer’s disease and 95.5% of patients with mild rheumatoid arthritis was higher than normal range. In addition, the antibody level was significantly higher in patients with rheumatoid arthritis (54.7±4.8) compared to patients with Alzheimer’s disease (P=0.0001).Conclusion: We obviously observed differences in Anti-CCP titer between patients with Alzheimer’s disease and those with mild rheumatoid arthritis. It was expected since anti-CCP has a high specificity for rheumatoid arthritis and become positive in early stages of disease with a high predictive value. However, we found that in early stages of cognitive impairment, we could not use this biomarker as a diagnostic test but during disease progression, a significant difference was observed between patients with mild cognitive impairment and those with Alzheimer’s disease. It needs more studies to confirm the role of anti-CCP antibody production in patients with Alzheimer’s disease.

We study -constaCyclic code over a semi-local ring with the condition , and , we show that -constaCyclic code over is equivalent to quasi-Cyclic code over by using two new Gray maps from to Also, for odd length we have defined a generating set for constaCyclic codes over Finally, we obtained some examples which are new to the data base [Database of codes [online]}, http:// codes.info(Accessed March 2, 2020)].