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02-06-2020 | Original Paper

# Multiple-rate error-correcting coding scheme

Authors: R. S. Raja Durai, Meenakshi Devi, Ashwini Kumar

## Abstract

Error-correcting codes that can effectively encode and decode messages of distinct lengths while maintaining a constant blocklength are considered. It is known conventionally that a k-dimensional block code of length n defined over $$\texttt {GF}(q^{n})$$ is designed to encode a k-symbol user data in to an n-length codeword, resulting in a fixed-rate coding. In contrast, considering $$q=p^{\lambda }$$, this paper proposes two coding procedures (for the cases of $$\lambda =k$$ and $$\lambda =n$$) each deriving a multiple-rate code from existing channel codes defined over a composite field $$\texttt {GF}(q^{n})$$. Formally, the proposed coding schemes employ $$\lambda$$ codes $${\mathcal {C}}_{1}(\lambda , 1), {\mathcal {C}}_{2}(\lambda , 2), \ldots , {\mathcal {C}}_{\lambda }(\lambda , \lambda )$$ defined over $$\texttt {GF}(q)$$ to encode user messages of distinct lengths and incorporate variable-rate feature. Unlike traditional block codes, the derived multiple-rate codes of fixed blocklength n can be used to encode and decode user messages $$\mathbf{m}$$ of distinct lengths $$|\mathbf{m}| = 1, 2, \ldots , k, k+1, \ldots , kn$$, thereby supporting a range of information rates—inclusive of the code rates $$1/n^{2}, 2/n^{2},\ldots , k/n^{2}$$ and $$1/n, 2/n, \ldots , k/n$$ ! A simple decoding procedure to the derived multiple-rate code is also given; in that, orthogonal projectors are employed for the identification of encoded user messages of variable length.
Literature
1.
Birkhoff, G., Mac Lane, S.: A Survey of Modern Algebra, 5th edn. Macmillan Publishers Ltd., New York (1996) MATH
2.
Tse, D., Viswanath, P.: Fundamentals of Wireless Communication. Cambridge University Press, Cambridge (2005) CrossRef
3.
Goldsmith, A., Chua, S.-G.: Adaptive coded modulation for fading channels. IEEE Trans. Commn. 46(5), 595–602 (1998) CrossRef
4.
Sun, Y., Karkooti, M., Cavallaro, J. R.: VLSI decoder architecture for high throughput, variable block-size and multi-rate LDPC codes. In: Proceedings of International Symposium on Circuits and Systems (ISCAS), pp. 2104–2107. New Orleans, LA (2007)
5.
Hagenauer, J.: Rate-compatible punctured convolutional codes and their application. IEEE Trans. Commun. 36, 389–400 (1988) CrossRef
6.
Acikel, O., Ryan, W.: Punctured turbo-codes for BPSK/QPSK channels. IEEE Trans. Commun. 47, 1315–1323 (1999) CrossRef
7.
Yazdani, M.R., Banihashemi, A.H.: On construction of rate-compatible low-density parity-check codes. IEEE Commun. Lett. 8(3), 159–161 (2004) CrossRef
8.
Ha, J., Kim, J., McLaughlin, S.: Rate-compatible puncturing of low-density parity-check codes. IEEE Trans. Inf. Theory 50(11), 2824–2836 (2004)
9.
Zhang, K., Ma, X., Zhao, S., Bai, B., Zhang X.: A new ensemble of rate-compatible LDPC codes. In: Proceedings of IEEE International Symposium on Information Theory (ISIT), pp. 2536–2540. Cambridge, MA (2012)
10.
Casado, A.I.V., Weng, W.-Y., Valle, S., Wesel, R.: Multiple-rate low-density parity-check codes with constant blocklength. IEEE Trans. Commun. 57(1), 75–83 (2009) CrossRef
11.
Liu, L., Zhou, W., Zhou, S.: Nonbinary multiple rate QC-LDPC codes with fixed information or block bit length. J. Commun. Netw. 14(4), 429–433 (2012) CrossRef
12.
Raja Durai, R. S.: Multiple-rate maximum rank distance codes. In: Proceedings of the 14th International Symposium on Information Theory and Its Applications (ISITA), pp. 696–699. Monterey, California (2016)
13.
Gabidulin, E.M.: Theory of codes with maximum rank distance. Probl. Inf. Transm. 21, 1–12 (1985)
14.
Lidl, R., Niederreiter, H.: Introduction to Finite Fields and Their Applications. Cambridge University Press, Cambridge (1986) MATH
15.
Hamming, R.W.: Error detecting and error correcting codes. Bell Syst. Techn. J. 29, 147–160 (1950)
16.
Roth, R.M.: Maximum-rank array codes and their application to crisscross error correction. IEEE Trans. Inf. Theory 37, 328–336 (1991)
17.
Delsarte, P.: Bilinear forms over a finite field, with applications to coding theory. J. Combin. Theory 25(3), 226–241 (1978)
18.
Cooperstein, B.N.: External flats to varieties in $${\cal{M}}_{n, n}$$( GF $$(q)$$). Linear Alg. Appl. 267, 175–186 (1997) MATH
19.
Loidreau, P.: Properties of codes in rank metric. In: Proceedings of the Eleventh International Workshop on Algebraic and Combinatorial Coding Theory, pp. 192–198. Bulgaria (2008)
20.
Gadouleau, M., Yan, Z.: On the decoder error probability of bounded rank-distance decoders for maximum rank distance codes. IEEE Trans. Inf. Theory 54(7), 3202–3206 (2008)
21.
Gadouleau, M., Yan, Z.: Packing and covering properties of rank metric codes. IEEE Trans. Inf. Theory 54(9), 3873–3883 (2008)
22.
Seroussi, G., Lempel, A.: Factorization of symmetric matrices and trace-orthogonal bases in finite fields. SIAM J. Comput. 9(4), 758–767 (1980)
23.
Lempel, A., Weinberger, M.: Self-complementary normal bases in finite fields. SIAM J. Discret. Math. 1, 193–198 (1988)
24.
Massey, J.L.: Linear codes with complementary duals. Discret. Math. 106 and 107, 337–342 (1992)
25.
Raja Durai, R. S.: On linear codes with rank metric: constructions, properties, and applications. Ph.D dissertation, Department of Mathematics, Indian Institute of Technology - Chennai, India (2004)
26.
Vasantha Kandasamy, W.B., Smarandache, F., Sujatha, R., Raja Durai, R.S.: Erasure Techniques in MRD Codes. ZIP Publishing, Columbus (2012) MATH
27.
Raja Durai, R.S., Devi, M.: On the class of T-Direct codes over GF $$(2^{{\cal{N}}})$$. Int. J. Comput. Inf. Syst. Ind. Manag. Appl. 5, 589–596 (2013)
28.
Liu, X., Liu, H.: Rank-metric complementary dual codes. J. Appl. Math. Comput. 61, 281–295 (2019)
29.
Green, D.H., Taylor, I.S.: Irreducible polynomials over composite Galois fields and their applications in coding techniques. Proc. Inst. Electr. Eng. 121, 935–939 (1974)
Title
Multiple-rate error-correcting coding scheme
Authors
R. S. Raja Durai
Meenakshi Devi
Ashwini Kumar
Publication date
02-06-2020
Publisher
Springer Berlin Heidelberg
Published in
Applicable Algebra in Engineering, Communication and Computing / Issue 2/2022
Print ISSN: 0938-1279
Electronic ISSN: 1432-0622
DOI
https://doi.org/10.1007/s00200-020-00435-x

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