Convolutional code[1] 

Description

Infinite-block code that is formed using generator polynomials over the finite field with two elements. The encoder slides across contiguous subsets of the input bit-string (like a convolutional neural network) evaluating the polynomials on that window to obtain a number of parity bits. These parity bits are the encoded information.

Rate

Depends on the polynomials used. Using puncturing removal [2] the rate for the code can be increased from \(\frac{1}{t}\) to \(\frac{s}{t}\), where \(t\) is the number of output bits, and \(s\) depends on the puncturing done. This is done by deleting some pieces of the encoder output such that the most-likely decoders remain effective

Encoding

Evaluation on the generator polynomials. Can be implemented with a small number of XOR gates

Decoding

Decoders based on the Viterbi algorithm (trellis decoding) were developed first, which result in the most-likely codeword for the encoded bits [3]. Following, other trellis decoders such as the BCJR decoding algorithm [4] were developed later.

Realizations

A type of convolutional code used in Real-time Application networks [5].Mobile and radio communications (3G networks) use convolutional codes concatenated with Reed-Solomon codes to obtain suitable performance [6].A convolutional code with rate 1/2 was used for deep-space and satellite communication [7]

Parent

  • Block code — Convolutional codes for infinite block size are block codes consisting of infinite blocks.

Children

Cousins

References

[1]
Peter Elias. Coding for noisy channels. IRE Convention Records, 3(4):37–46, 1955.
[2]
L. Sari, “Effects of Puncturing Patterns on Punctured Convolutional Codes”, TELKOMNIKA (Telecommunication, Computing, Electronics and Control) 10, (2012) DOI
[3]
A. Viterbi, “Error bounds for convolutional codes and an asymptotically optimum decoding algorithm”, IEEE Transactions on Information Theory 13, 260 (1967) DOI
[4]
L. Bahl et al., “Optimal decoding of linear codes for minimizing symbol error rate (Corresp.)”, IEEE Transactions on Information Theory 20, 284 (1974) DOI
[5]
S. I. Mrutu, A. Sam, and N. H. Mvungi, “Forward Error Correction Convolutional Codes for RTAs’ Networks: An Overview”, International Journal of Computer Network and Information Security 6, 19 (2014) DOI
[6]
T. Halonen, J. Romero, and J. Melero, editors , GSM, GPRS and EDGE Performance (Wiley, 2003) DOI
[7]
Butman, Deutsch, and Miller. Performance of concatenated codes for deep space missions. 1981.
[8]
G. D. Forney, Jr., “Why quasi cyclic codes are interesting,” unpublished note, 1970.
[9]
G. Solomon and H. C. A. Tilborg, “A Connection Between Block and Convolutional Codes”, SIAM Journal on Applied Mathematics 37, 358 (1979) DOI
[10]
R. Michael Tanner, “Error-correcting coding system,” U.S. Patent 4295218, 1981.
[11]
R. Michael Tanner. Convolutional codes from quasi-cyclic codes: A link between the theories of block and convolutional codes. University of California, Santa Cruz, Computer Research Laboratory, 1987.
[12]
“Generalized tail-biting convolutional codes,” Ph.D. dissertation, Univ. of Massachusetts, Amherst, 1985.
[13]
Y. Levy and J. Costello, Jr., “An algebraic approach to constructing convolutional codes from quasi-cyclic codes,” DIMACS Ser. Discr. Math. and Theor. Comp. Sci., vol. 14, pp. 189–198, 1993.
[14]
M. Esmaeili et al., “A link between quasi-cyclic codes and convolutional codes”, IEEE Transactions on Information Theory 44, 431 (1998) DOI
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Zoo Code ID: convolutional

Cite as:
“Convolutional code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2021. https://errorcorrectionzoo.org/c/convolutional
BibTeX:
@incollection{eczoo_convolutional,
  title={Convolutional code},
  booktitle={The Error Correction Zoo},
  year={2021},
  editor={Albert, Victor V. and Faist, Philippe},
  url={https://errorcorrectionzoo.org/c/convolutional}
}
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Cite as:

“Convolutional code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2021. https://errorcorrectionzoo.org/c/convolutional

Github: https://github.com/errorcorrectionzoo/eczoo_data/tree/main/codes/classical/q-ary_digits/convolutional/convolutional.yml.