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Problems of Information Transmission

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01.01.2021 | CODING THEORY

On the Generalized Concatenated Construction for Codes in \({L_1}\) and Lee Metrics

verfasst von: V. A. Zinoviev, D. V. Zinoviev

Erschienen in: Problems of Information Transmission | Ausgabe 1/2021

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Abstract

We consider a generalized concatenated construction for error-correcting codes over the q-ary alphabet in the modulus metric L₁ and Lee metric L. Resulting codes have arbitrary length, arbitrary distance (independently of the alphabet size), and can correct both independent errors and error bursts in both metrics. In particular, for any length 2^m we construct codes over \(\mathbb{Z}_4\) with Lee distance 4 which under the Gray mapping yield extended binary perfect codes of length 2^m+1 (with code distance 4). We construct codes over \(\mathbb{Z}_4\) of length n with Lee distance n which under the Gray mapping yield Hadamard matrices of order 2n (under the additional condition that an Hadamard matrix of order n exists). The constructed new codes in the Lee metric are often better in their parameters than previously known ones; in particular, they are essentially better than previously constructed Astola codes.

Vorheriger Artikel The f-Divergence and Coupling of Probability Distributions

Nächster Artikel Affine Variety Codes over a Hyperelliptic Curve

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Zinoviev, D.V. and Zinoviev, V.A., On Generalized Concatenated Construction of Codes in Metrics Lee L and L₁, in Proc. 16th Int. Workshop on Algebraic and Combinatorial Coding Theory (ACCT’2018), Svetlogorsk (Kaliningrad region), Russia, Sept. 2–8, 2018, pp. 62–65. Available at https://www.dropbox.com/s/h7u89lh8vyirww9.

Berlekamp, E.R., Negacyclic Codes for the Lee Metric, Proc. Conf. on Combinatorial Mathematics and Its Applications, Chapel Hill, NC, Apr. 10–14, 1967, Bose, R.C. and Dowling, T.A., Eds., Chapel Hill: Univ. of North Carolina Press, 1968, ch. 17, pp. 298–316. Reprinted in: Berlekamp, E.R., Algebraic Coding Theory, Rev. Ed., Singapore: World Sci., 2015, ch. 9, pp. 207–217. https://doi.org/10.1142/9789814635905_0009

Chiang, J.C. and Wolf, J.K., On Channels and Codes for the Lee Metric, Inform. Control, 1971, vol. 19, no. 2, pp. 159–174. https://doi.org/10.1016/S0019-9958(71)90791-1MathSciNetCrossRef

Mazur, L.E., Codes Correcting Errors of Large Weight in Lee Metric, Probl. Peredachi Inf., 1973, vol. 9, no. 4, pp. 11–16 [Probl. Inf. Transm. (Engl. Transl.), 1973, vol. 9, no. 4, pp. 277–281]. http://mi.mathnet.ru/eng/ppi917MathSciNetMATH

Astola, J., Concatenated Codes for the Lee Metric, IEEE Trans. Inform. Theory, 1982, vol. 28, no. 5, pp. 778–779. https://doi.org/10.1109/TIT.1982.1056550MathSciNetCrossRef

Racsmáany, A., On Constructing Codes with Given Distance in Lee-Metric, Probl. Control Inform. Theory, 1986, vol. 15, no. 5, pp. 377–384.MathSciNet

Davydov, V.A., Codes Correcting Errors in the Modulus Metric, Lee Metric, and Operator Errors, Probl. Peredachi Inf., 1993, vol. 29, no. 3, pp. 10–20 [Probl. Inf. Transm. (Engl. Transl.), 1993, vol. 29, no. 3, pp. 209–217]. http://mi.mathnet.ru/eng/ppi184MATH

Davydov, V.A., On Application of the Modulus Metric to Solving the Minimum Euclidean Distance Decoding Problem, Probl. Peredachi Inf., 2019, vol. 55, no. 2, pp. 50–57 [Probl. Inf. Transm. (Engl. Transl.), 2019, vol. 55, no. 2, pp. 145–151]. https://doi.org/10.1134/S0032946019020030MathSciNet

Ericson, T. and Zinoviev, V., Spherical Codes Generated by Binary Partitions of Symmetric Pointsets, Probl. Peredachi Inf., IEEE Trans. Inform. Theory, 1995, vol. 41, no. 1, pp. 107–129. https://doi.org/10.1109/18.370114MathSciNetCrossRef

10.

Zinoviev, V.A., Generalized Concatenated Codes, Probl. Peredachi Inf., Probl. Peredachi Inf., 1976, vol. 12, no. 1, pp. 5–15 [Probl. Inf. Transm. (Engl. Transl.), 1976, vol. 12, no. 1, pp. 2–9]. http://mi.mathnet.ru/eng/ppi1670

11.

Dumer, I., Zinoviev, V., and Zyablov, V., Concatenated Decoding According to Minimal Generalized Distance, Probl. Control Inform. Theory, 1981, vol. 10, no. 1, pp. 3–19.

12.

Zinoviev, V.A. and Zyablov, V.V., Correction of Error Bursts and Independent Errors using Generalized Concatenated Codes, Probl. Peredachi Inf., 1979, vol. 15, no. 2, pp. 58–70 [Probl. Inf. Transm. (Engl. Transl.), 1979, vol. 15, no. 2, pp. 125–134]. 1976, vol. 12, no. 1, pp. 5–15 [Probl. Inf. Transm. (Engl. Transl.), 1976, vol. 12, no. 1, pp. 2–9]. http://mi.mathnet.ru/eng/ppi1488

13.

Zinoviev, V.A. and Zyablov, V.V., Codes with Unequal Protection of Information Symbols, Probl. Peredachi Inf., 1979, vol. 15, no. 2, pp. 58–70 [Probl. Inf. Transm. (Engl. Transl.), 1979, vol. 15, no. 2, pp. 125–134]. 1976, vol. 12, no. 1, pp. 5–15 [Probl. Inf. Transm. (Engl. Transl.), 1976, vol. 12, no. 1, pp. 2–9]. http://mi.mathnet.ru/eng/ppi1499MathSciNetMATH

14.

Hammons, A.R., Kumar, P.V., Calderbank, A.R., Sloane, N.J.A., and Solé, P., The Z4-Linearity of Kerdock, Preparata, Goethals and Related Codes, IEEE Trans. Inform. Theory, 1994, vol. 40, no. 2, pp. 301–319. https://doi.org/10.1109/18.312154MathSciNetCrossRef

15.

Krotov, D.S., Z4-Linear Hadamard and Extended Perfect Codes, Electron. Notes Discrete Math., 2001, vol. 6, pp. 107–112. https://doi.org/10.1016/S1571-0653(04)00161-1MathSciNetCrossRef

16.

Vasil’ev, Yu.L., On Nongroup Densely Packed Codes, Probl. Kibern., 1962, vol. 8, pp. 337–339.MATH

17.

Krotov, D.S. and Villanueva, M., Classification of the \(\mathbb{Z}_2\mathbb{Z}_4\)-Linear Hadamard Codes and Their Automorphism Groups, IEEE Trans. Inform. Theory, 2015, vol. 61, no. 2, pp. 887–894. https://doi.org/10.1109/TIT.2014.2379644MathSciNetCrossRef

18.

Grassl, M., Bounds on the Minimum Distance of Linear Codes and Quantum Codes (electronic tables). Available online at http://www.codetables.de (accessed on Oct. 9, 2018).

Titel: On the Generalized Concatenated Construction for Codes in and Lee Metrics
verfasst von: V. A. Zinoviev
D. V. Zinoviev
Publikationsdatum: 01.01.2021
Verlag: Pleiades Publishing
Erschienen in: Problems of Information Transmission / Ausgabe 1/2021
Print ISSN: 0032-9460
Elektronische ISSN: 1608-3253
DOI: https://doi.org/10.1134/S003294602101004X

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