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Some algebraic and distance properties of convolutional codes.

James L. Massey

Published Jan 1, 1968• 3 citations
OpenAlex

Abstract

Upper and lower bounds for feedback decoding and definite decoding minimum distances of binary convolutional codes

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Error propagation and definite decoding of convolutional codes

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The error-propagation effect in decoding convolutional codes is a result of the internal feedback in the usual decoding method, feedback decoding (FD). As a measure of this effect, the propagation lengthLof a system is defined as the maximum span of decoding errors following a decoding error when all succeding parity checks are satisfied. A relationship betweenL, the parity check matrix, and the decoding algorithm is developed. A decoding method having no internal feedback, definite decoding (DD), is formalized. It is shown that a code using FD with limitedLexists if and only if that same code can be decoded using DD. When using DD a smaller class of errors is corrected. The self-orthogonal codes are shown to be decodable using FD withLsmall. The minimum possible value ofLwhen using bounded distance decoding is given for some of these codes. Codes are given which minimize the spacing between single correctable errors using DD. These values of spacing are compared with those for similar (known) codes which use FD, and with the theoretical minimum spacing.

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A lower bound on the minimum distance of convolutional polar codes is provided. The bound is obtained from the minimum weight of the generalized cosets of the codes generated by the bottom rows of the polarizing matrix. Moreover, a construction of convolutional polar subcodes is proposed, which provides improved performance under successive cancellation list decoding. For sufficiently large list size, the decoding complexity of convolutional polar subcodes appears to be lower compared with Arikan polar subcodes with the same performance. The error probability of successive cancellation list decoding of convolutional polar subcodes is lower than that of Arikan polar subcodes with the same list size.

arXiv:1901.06341•IEEE Transactions on Communications•Apr 4, 2019•9 citations

Asymptotic distance and convergence analysis of braided protograph convolutional codes

Marcos B. S. Tavares, Michael Lentmaier, Gerhard Paul Fettweis +1 more

We analyze a class of LDPC convolutional codes that are constructed from tightly braided convolutional base codes by a lifting procedure. For these braided protograph convolutional codes, we show that the distances grow linearly with the constraint length, and we present lower bounds on their asymptotic segment distance and free distance as well as on the asymptotic minimum distance of their tail-biting versions. With some constraints imposed on the lifting permutations, braided protograph convolutional codes can also be decoded as turbo-like codes by iterative application of the BCJR algorithm. For this case, we derive an explicit upper-bound on the asymptotic decoding error probability as a function of the number of iterations.

Allerton Conference on Communication, Control, and Computing•Sep 1, 2008•5 citations

New Tight Bounds on Error Probabilities of Binary Convolutional Codes

Hung Nguyen Tung, Takeshi Hashimoto

IEEE International Symposium on Information Theory (ISIT)•Dec 15, 2003•0 citations
The error-propagation effect in decoding convolutional codes is a result of the internal feedback in the usual decoding method, feedback decoding (FD). As a measure of this effect, the propagation lengthLof a system is defined as the maximum span of decoding errors following a decoding error when all succeding parity checks are satisfied. A relationship betweenL, the parity check matrix, and the decoding algorithm is developed. A decoding method having no internal feedback, definite decoding (DD), is formalized. It is shown that a code using FD with limitedLexists if and only if that same code can be decoded using DD. When using DD a smaller class of errors is corrected. The self-orthogonal codes are shown to be decodable using FD withLsmall. The minimum possible value ofLwhen using bounded distance decoding is given for some of these codes. Codes are given which minimize the spacing between single correctable errors using DD. These values of spacing are compared with those for similar (known) codes which use FD, and with the theoretical minimum spacing.
A lower bound on the minimum distance of convolutional polar codes is provided. The bound is obtained from the minimum weight of the generalized cosets of the codes generated by the bottom rows of the polarizing matrix. Moreover, a construction of convolutional polar subcodes is proposed, which provides improved performance under successive cancellation list decoding. For sufficiently large list size, the decoding complexity of convolutional polar subcodes appears to be lower compared with Arikan polar subcodes with the same performance. The error probability of successive cancellation list decoding of convolutional polar subcodes is lower than that of Arikan polar subcodes with the same list size.
We analyze a class of LDPC convolutional codes that are constructed from tightly braided convolutional base codes by a lifting procedure. For these braided protograph convolutional codes, we show that the distances grow linearly with the constraint length, and we present lower bounds on their asymptotic segment distance and free distance as well as on the asymptotic minimum distance of their tail-biting versions. With some constraints imposed on the lifting permutations, braided protograph convolutional codes can also be decoded as turbo-like codes by iterative application of the BCJR algorithm. For this case, we derive an explicit upper-bound on the asymptotic decoding error probability as a function of the number of iterations.