This paper provides a construction of non-binary LDPC convolutional codes, which generalizes the work of Robinson and Bernstein. The sets of integers forming an (n - 1,w)- difference triangle set are used as supports of the columns of rate (n - 1)/n convolutional codes. If the field size is large enough, the Tanner graph associated to the sliding parity-check matrix of the code is free from 4 and 6-cycles not satisfying the full rank condition. This is important for improving the performance of a code and avoiding the presence of low-weight codewords and absorbing sets. The parameters of the convolutional code are shown to be determined by the parameters of the underlying difference triangle set. In particular, the free distance of the code is related to w and the degree of the code is linked to the "scope" of the difference triangle set. Hence, the problem of finding families of difference triangle set with minimum scope is equivalent to find convolutional codes with small degree.This paper provides a construction of non-binary LDPC convolutional codes, which generalizes the work of Robinson and Bernstein. The sets of integers forming an (n - 1,w)- difference triangle set are used as supports of the columns of rate (n - 1)/n convolutional codes. If the field size is large enough, the Tanner graph associated to the sliding parity-check matrix of the code is free from 4 and 6-cycles not satisfying the full rank condition. This is important for improving the performance of a code and avoiding the presence of low-weight codewords and absorbing sets. The parameters of the convolutional code are shown to be determined by the parameters of the underlying difference triangle set. In particular, the free distance of the code is related to w and the degree of the code is linked to the "scope" of the difference triangle set. Hence, the problem of finding families of difference triangle set with minimum scope is equivalent to find convolutional codes with small degree.

arXiv:2001.07969IEEE International Symposium on Information Theory (ISIT)Jun 1, 20202 citations

Constructing Quantum Convolutional Codes via Difference Triangle Sets

Vahid Nourozi, David Mitchell

In this paper, we introduce a construction of quantum convolutional codes (QCCs) based on difference triangle sets (DTSs). To construct QCCs, one must determine polynomial stabilizers $X(D)$ and $Z(D)$ that commute (symplectic orthogonality), while keeping the stabilizers sparse and encoding memory small. To construct Z(D), we show that one can use a reflection of the DTS indices of X(D), where X(D) corresponds to a classical convolutional self-orthogonal code (CSOC) constructed from strong DTS supports. The motivation of this approach is to provide a constructive design that guarantees a prescribed minimum distance. We provide numerical results demonstrating the construction for a variety of code rates.In this paper, we introduce a construction of quantum convolutional codes (QCCs) based on difference triangle sets (DTSs). To construct QCCs, one must determine polynomial stabilizers $X(D)$ and $Z(D)$ that commute (symplectic orthogonality), while keeping the stabilizers sparse and encoding memory small. To construct Z(D), we show that one can use a reflection of the DTS indices of X(D), where X(D) corresponds to a classical convolutional self-orthogonal code (CSOC) constructed from strong DTS supports. The motivation of this approach is to provide a constructive design that guarantees a prescribed minimum distance. We provide numerical results demonstrating the construction for a variety of code rates.

arXiv:2602.13505Feb 13, 20260 citations

Asymmetric quantum convolutional codes

Giuliano G. La Guardia

Quantum Information ProcessingNov 20, 20153 citations

Asymmetric quantum convolutional codes

Giuliano G. La Guardia

In this paper, we construct the first families of asymmetric quantum convolutional codes (AQCC)'s. These new AQCC's are constructed by means of the CSS-type construction applied to suitable families of classical convolutional codes, which are also constructed here. The new codes have noncatastrophic generator matrices and they present great asymmetry. Since our constructions are performed algebraically, it is possible to derive several families of such codes and not only codes with specific parameters. Additionally, several different types of such codes are obtained.In this paper, we construct the first families of asymmetric quantum convolutional codes (AQCC)'s. These new AQCC's are constructed by means of the CSS-type construction applied to suitable families of classical convolutional codes, which are also constructed here. The new codes have noncatastrophic generator matrices and they present great asymmetry. Since our constructions are performed algebraically, it is possible to derive several families of such codes and not only codes with specific parameters. Additionally, several different types of such codes are obtained.

arXiv:1504.05639Apr 22, 20150 citations

In this paper, we introduce a construction of quantum convolutional codes (QCCs) based on difference triangle sets (DTSs). To construct QCCs, one must determine polynomial stabilizers

X(D)

and

Z(D)

that commute (symplectic orthogonality), while keeping the stabilizers sparse and encoding memory small. To construct Z(D), we show that one can use a reflection of the DTS indices of X(D), where X(D) corresponds to a classical convolutional self-orthogonal code (CSOC) constructed from strong DTS supports. The motivation of this approach is to provide a constructive design that guarantees a prescribed minimum distance. We provide numerical results demonstrating the construction for a variety of code rates.